3214:
2602:
5300:
8703:
3209:{\displaystyle {\begin{aligned}&\Delta t\sum _{i=1}^{N}\left(\left|\int _{0}^{1}f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta \right|-\int _{0}^{1}\left|f'(t_{i})\right|d\theta \right)\\&\qquad \leqq \Delta t\sum _{i=1}^{N}\left(\int _{0}^{1}\left|f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|\ d\theta -\int _{0}^{1}\left|f'(t_{i})\right|d\theta \right)\\&\qquad =\Delta t\sum _{i=1}^{N}\int _{0}^{1}\left|f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|-\left|f'(t_{i})\right|\ d\theta \end{aligned}}}
4921:
10233:
7606:
39:
8388:
5316:
10728:
10720:
164:
5295:{\displaystyle {\begin{aligned}L(f)&=\int _{a}^{b}{\Big |}f'(t){\Big |}\ dt=\int _{a}^{b}{\Big |}g'(\varphi (t))\varphi '(t){\Big |}\ dt\\&=\int _{a}^{b}{\Big |}g'(\varphi (t)){\Big |}\varphi '(t)\ dt\quad {\text{in the case }}\varphi {\text{ is non-decreasing}}\\&=\int _{c}^{d}{\Big |}g'(u){\Big |}\ du\quad {\text{using integration by substitution}}\\&=L(g).\end{aligned}}}
7347:
7860:
1080:
6847:
8698:{\displaystyle \left(\mathbf {x} _{r}\cdot \mathbf {x} _{r}\right)\left(r'^{2}\right)+\left(\mathbf {x} _{\theta }\cdot \mathbf {x} _{\theta }\right)\left(\theta '\right)^{2}+\left(\mathbf {x} _{\phi }\cdot \mathbf {x} _{\phi }\right)\left(\phi '\right)^{2}=\left(r'\right)^{2}+r^{2}\left(\theta '\right)^{2}+r^{2}\sin ^{2}\theta \left(\phi '\right)^{2}.}
8912:
3782:
7613:
10638:
779:
11519:
2593:
9095:
6617:
10220:
In 1659 van
Heuraet published a construction showing that the problem of determining arc length could be transformed into the problem of determining the area under a curve (i.e., an integral). As an example of his method, he determined the arc length of a semicubical parabola, which required finding
10137:
within the curves and compute the length of the sides for a somewhat accurate measurement of the length. By using more segments, and by decreasing the length of each segment, they were able to obtain a more and more accurate approximation. In particular, by inscribing a polygon of many sides in a
11633:
The sign in the square root is chosen once for a given curve, to ensure that the square root is a real number. The positive sign is chosen for spacelike curves; in a pseudo-Riemannian manifold, the negative sign may be chosen for timelike curves. Thus the length of a curve is a non-negative real
7601:{\displaystyle \left(\mathbf {x_{r}} \cdot \mathbf {x_{r}} \right)\left(r'\right)^{2}+2\left(\mathbf {x} _{r}\cdot \mathbf {x} _{\theta }\right)r'\theta '+\left(\mathbf {x} _{\theta }\cdot \mathbf {x} _{\theta }\right)\left(\theta '\right)^{2}=\left(r'\right)^{2}+r^{2}\left(\theta '\right)^{2}.}
10047:
Those definitions of the metre and the nautical mile have been superseded by more precise ones, but the original definitions are still accurate enough for conceptual purposes and some calculations. For example, they imply that one kilometre is exactly 0.54 nautical miles. Using official modern
5751:
is necessary. Numerical integration of the arc length integral is usually very efficient. For example, consider the problem of finding the length of a quarter of the unit circle by numerically integrating the arc length integral. The upper half of the unit circle can be parameterized as
1321:
2372:
8296:
4304:
222:
If the curve is not already a polygonal path, then using a progressively larger number of line segments of smaller lengths will result in better curve length approximations. Such a curve length determination by approximating the curve as connected (straight) line segments is called
8710:
2071:
3476:
1754:
6626:
3576:
8183:
7340:
8919:
10474:
2382:
11307:
567:
7855:{\displaystyle \int _{t_{1}}^{t_{2}}{\sqrt {\left({\frac {dr}{dt}}\right)^{2}+r^{2}\left({\frac {d\theta }{dt}}\right)^{2}\,}}dt=\int _{\theta (t_{1})}^{\theta (t_{2})}{\sqrt {\left({\frac {dr}{d\theta }}\right)^{2}+r^{2}\,}}d\theta .}
3893:
5594:
3325:
11275:
6125:
1075:{\displaystyle L(f)=\lim _{N\to \infty }\sum _{i=1}^{N}{\bigg |}f(t_{i})-f(t_{i-1}){\bigg |}=\lim _{N\to \infty }\sum _{i=1}^{N}\left|{\frac {f(t_{i})-f(t_{i-1})}{\Delta t}}\right|\Delta t=\int _{a}^{b}{\Big |}f'(t){\Big |}\ dt.}
6466:
6234:
7252:
6964:
6455:
5691:
2197:
4602:
1094:
7183:
11141:
4698:
4132:
1575:
9706:
6030:
227:
of a curve. The lengths of the successive approximations will not decrease and may keep increasing indefinitely, but for smooth curves they will tend to a finite limit as the lengths of the segments get
4891:
10700:
8008:
1580:
10431:
9754:
9577:
3943:
725:
8343:
8190:
8073:
11585:
10784:. Informally, such curves are said to have infinite length. There are continuous curves on which every arc (other than a single-point arc) has infinite length. An example of such a curve is the
10479:
9625:
4926:
2607:
7105:
3569:
1920:
3998:
1401:
8907:{\displaystyle \int _{t_{1}}^{t_{2}}{\sqrt {\left({\frac {dr}{dt}}\right)^{2}+r^{2}\left({\frac {d\theta }{dt}}\right)^{2}+r^{2}\sin ^{2}\theta \left({\frac {d\phi }{dt}}\right)^{2}\,}}dt.}
5935:
3531:
649:
371:
142:
10344:
6388:
9845:
4061:
1925:
11069:
4352:
4444:
3330:
2126:
10914:
1862:
10280:
5796:
2192:
1789:
442:
6321:
5491:
4087:
31:
10775:
10633:{\displaystyle {\begin{aligned}AC^{2}&=AB^{2}+BC^{2}\\&=\varepsilon ^{2}+{9 \over 4}a\varepsilon ^{2}\\&=\varepsilon ^{2}\left(1+{9 \over 4}a\right)\end{aligned}}}
6842:{\displaystyle \left(\mathbf {x} _{u}u'+\mathbf {x} _{v}v'\right)\cdot (\mathbf {x} _{u}u'+\mathbf {x} _{v}v')=g_{11}\left(u'\right)^{2}+2g_{12}u'v'+g_{22}\left(v'\right)^{2}}
5354:
772:
6037:
4812:
4027:
7897:
6154:
1882:
11514:{\displaystyle \ell (\gamma )=\int \limits _{0}^{1}{\sqrt {\pm \sum _{i,j=1}^{n}g_{ij}(x(\gamma (t))){\frac {dx^{i}(\gamma (t))}{dt}}{\frac {dx^{j}(\gamma (t))}{dt}}}}dt}
9499:
7257:
3777:{\displaystyle \Delta t\sum _{i=1}^{N}\left(\left|\int _{0}^{1}f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta \right|-\left|f'(t_{i})\right|\right)<\varepsilon N\Delta t}
2164:
1439:
5865:
9916:
9343:
7923:
9438:
9375:
5603:
11608:
11166:
10020:
9960:
9776:
9647:
9522:
9463:
9246:
8028:
7030:
6997:
5480:
5392:
9285:
6879:
9803:
9399:
9090:{\displaystyle \int _{t_{1}}^{t_{2}}{\sqrt {\left({\frac {dr}{dt}}\right)^{2}+r^{2}\left({\frac {d\theta }{dt}}\right)^{2}+\left({\frac {dz}{dt}}\right)^{2}\,}}dt.}
8068:
4490:
10125:", few believed it was even possible for curves to have definite lengths, as do straight lines. The first ground was broken in this field, as it often has been in
4107:
5442:
11631:
4914:
11299:
10954:
10934:
10866:
10842:
10000:
9940:
9305:
9226:
9202:
9178:
9150:
8383:
8363:
8048:
5412:
4753:
4733:
4387:
4127:
399:
281:
253:
4476:
1821:
1491:
435:
12110:
7112:
2588:{\displaystyle \sum _{i=1}^{N}\left|\int _{0}^{1}f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta \right|\Delta t-\sum _{i=1}^{N}\left|f'(t_{i})\right|\Delta t.}
13056:
10780:
As mentioned above, some curves are non-rectifiable. That is, there is no upper bound on the lengths of polygonal approximations; the length can be made
11174:
3789:
13044:
5310:
3221:
7190:
6393:
6888:
6612:{\displaystyle D(\mathbf {x} \circ \mathbf {C} )=(\mathbf {x} _{u}\ \mathbf {x} _{v}){\binom {u'}{v'}}=\mathbf {x} _{u}u'+\mathbf {x} _{v}v'.}
12049:
13166:
13051:
2367:{\displaystyle \sum _{i=1}^{N}\left|{\frac {f(t_{i})-f(t_{i-1})}{\Delta t}}\right|\Delta t-\sum _{i=1}^{N}\left|f'(t_{i})\right|\Delta t.}
13207:
11084:
4299:{\displaystyle \sum _{i=1}^{N}\left|{\frac {f(t_{i})-f(t_{i-1})}{\Delta t}}\right|\Delta t=\sum _{i=1}^{N}\left|f'(t_{i})\right|\Delta t}
13034:
13029:
10209:
Before the full formal development of calculus, the basis for the modern integral form for arc length was independently discovered by
4613:
1316:{\displaystyle f(t_{i})-f(t_{i-1})=\int _{t_{i-1}}^{t_{i}}f'(t)\ dt=\Delta t\int _{0}^{1}f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta }
13039:
13024:
12138:
1496:
12326:
9660:
5940:
13019:
10058:
nautical miles. This modern ratio differs from the one calculated from the original definitions by less than one part in 10,000.
4821:
12006:
Farouki, Rida T. (1999). "Curves from motion, motion from curves". In
Laurent, P.-J.; Sablonniere, P.; Schumaker, L. L. (eds.).
10649:
7931:
8291:{\displaystyle D(\mathbf {x} \circ \mathbf {C} )=\mathbf {x} _{r}r'+\mathbf {x} _{\theta }\theta '+\mathbf {x} _{\phi }\phi '.}
10355:
9711:
12636:
12390:
12015:
11857:
11823:
9531:
7107:
be a curve expressed in polar coordinates. The mapping that transforms from polar coordinates to rectangular coordinates is
4485:
The definition of arc length of a smooth curve as the integral of the norm of the derivative is equivalent to the definition
3898:
654:
8304:
6130:
9892:
on the Earth's surface would be simply numerically related to the angles they subtend at its centre. The simple equation
9582:
12188:
12104:
12081:
7043:
3536:
1887:
3948:
1326:
13134:
12993:
11749:
17:
11531:
12548:
12464:
5870:
2066:{\displaystyle \left|\left|f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|-\left|f'(t_{i})\right|\right|<\varepsilon }
3481:
1749:{\displaystyle {\frac {f(t_{i})-f(t_{i-1})}{\Delta t}}=\int _{0}^{1}f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta .}
574:
323:
94:
13129:
13061:
12686:
12541:
12509:
12268:
10307:
6326:
1088:
9808:
4031:
3471:{\textstyle \left|\left|f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|-\left|f'(t_{i})\right|\right|<\varepsilon }
12762:
12739:
12454:
8178:{\displaystyle \mathbf {x} (r,\theta ,\phi )=(r\sin \theta \cos \phi ,r\sin \theta \sin \phi ,r\cos \theta ).}
8070:
is the azimuthal angle. The mapping that transforms from spherical coordinates to rectangular coordinates is
255:
that is an upper bound on the length of all polygonal approximations (rectification). These curves are called
12852:
12790:
12585:
12459:
12131:
12042:
10964:
4396:
2076:
68:
Determining the length of an irregular arc segment by approximating the arc segment as connected (straight)
12338:
12316:
10875:
10781:
1838:
13161:
10246:
10117:, even the greatest thinkers considered it impossible to compute the length of an irregular arc. Although
5755:
13146:
12912:
12526:
12348:
12087:
12037:
10150:
In the 17th century, the method of exhaustion led to the rectification by geometrical methods of several
5415:
2169:
378:
88:
8916:
A very similar calculation shows that the arc length of a curve expressed in cylindrical coordinates is
4313:
13192:
12531:
12301:
12032:
11870:
Richeson, David (May 2015). "Circular
Reasoning: Who First Proved That C Divided by d Is a Constant?".
10845:
10139:
6289:
4066:
34:
When rectified, the curve gives a straight line segment with the same length as the curve's arc length.
10734:
5330:
730:
12950:
12897:
9865:
4916:
The arc length of the curve is the same regardless of the parameterization used to define the curve:
1761:
291:
12358:
10048:
definitions, one nautical mile is exactly 1.852 kilometres, which implies that 1 kilometre is about
7335:{\displaystyle D(\mathbf {x} \circ \mathbf {C} )=\mathbf {x} _{r}r'+\mathbf {x} _{\theta }\theta '.}
13066:
12837:
12385:
12124:
11815:
5747:
In most cases, including even simple curves, there are no closed-form solutions for arc length and
4761:
4003:
1828:
10067:
7867:
1867:
12832:
12504:
12092:
9472:
5696:
4390:
2131:
1406:
5801:
12960:
12842:
12663:
12611:
12417:
12395:
12263:
11741:
11707:
10159:
10114:
9895:
9313:
7902:
6882:
152:
148:
11969:
9411:
9348:
13086:
12945:
12857:
12514:
12449:
12422:
12412:
12333:
12321:
12306:
12278:
11692:
11593:
11151:
10044:
nautical miles. Those are the numbers of the corresponding angle units in one complete turn.
10005:
9945:
9859:
9761:
9632:
9507:
9448:
9231:
8013:
7002:
6969:
5748:
5447:
5359:
562:{\displaystyle L(f)=\lim _{N\to \infty }\sum _{i=1}^{N}{\bigg |}f(t_{i})-f(t_{i-1}){\bigg |}}
381:(i.e., the derivative is a continuous function) function. The length of the curve defined by
12100:
11986:
11952:
11807:
10092:
9255:
6854:
12902:
12521:
12368:
11638:
10194:
10151:
10122:
10030:
The lengths of the distance units were chosen to make the circumference of the Earth equal
9785:
9381:
8053:
5988:
5902:
5720:
4893:
is another continuously differentiable parameterization of the curve originally defined by
4607:
295:
10788:. Another example of a curve with infinite length is the graph of the function defined by
5589:{\displaystyle {\sqrt {dx^{2}+dy^{2}}}={\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}\,}}dx.}
4092:
8:
12922:
12847:
12734:
12691:
12442:
12427:
12258:
12246:
12233:
12193:
12173:
11808:
11634:
number. Usually no curves are considered which are partly spacelike and partly timelike.
10809:
10465:
10210:
10197:. The accompanying figures appear on page 145. On page 91, William Neile is mentioned as
6247:
5421:
402:
204:
11968:
van
Heuraet, Hendrik (1659). "Epistola de transmutatione curvarum linearum in rectas ".
11613:
10097:
4896:
207:
in
Euclidean space, for example), the total length of the approximation can be found by
13011:
12986:
12817:
12770:
12711:
12676:
12671:
12651:
12646:
12641:
12606:
12553:
12536:
12437:
12311:
12296:
12241:
12208:
11933:
11895:
11734:
11687:
11284:
10939:
10919:
10851:
10827:
10155:
9985:
9925:
9871:
9290:
9211:
9187:
9163:
9135:
8368:
8348:
8033:
5724:
5712:
5483:
5397:
4738:
4703:
4357:
4112:
1832:
1824:
774:
This definition is equivalent to the standard definition of arc length as an integral:
384:
266:
238:
47:
11840:
10705:
In order to approximate the length, Fermat would sum up a sequence of short segments.
6120:{\displaystyle \int _{-{\sqrt {2}}/2}^{{\sqrt {2}}/2}{\frac {dx}{\sqrt {1-x^{2}}}}\,.}
4449:
1794:
1444:
408:
13151:
12975:
12907:
12729:
12706:
12580:
12573:
12476:
12291:
12183:
12056:
12011:
11899:
11887:
11819:
11788:
11745:
11677:
10813:
10714:
10225:. In 1660, Fermat published a more general theory containing the same result in his
10179:
9249:
6229:{\displaystyle \arcsin x{\bigg |}_{-{\sqrt {2}}/2}^{{\sqrt {2}}/2}={\frac {\pi }{2}}}
5736:
299:
229:
80:
these approximations don't get arbitrarily large (so the curve has a finite length).
11883:
9116:
4735:
This definition as the supremum of the all possible partition sums is also valid if
13109:
12892:
12805:
12785:
12716:
12626:
12568:
12560:
12494:
12407:
12168:
12163:
11925:
11879:
11778:
10214:
10171:
4307:
214:
188:
11270:{\displaystyle \ell (\gamma )=\int \limits _{0}^{1}||\gamma '(t)||_{\gamma (t)}dt}
3888:{\displaystyle \left|f'(t_{i})\right|=\int _{0}^{1}\left|f'(t_{i})\right|d\theta }
13202:
13197:
13171:
13156:
12940:
12795:
12775:
12744:
12721:
12701:
12595:
12251:
12198:
11913:
11672:
11650:
10190:
9779:
6275:
5686:{\displaystyle s=\int _{a}^{b}{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}\,}}dx.}
303:
196:
180:
4389:
This definition of arc length shows that the length of a curve represented by a
38:
13081:
12980:
12827:
12780:
12681:
12484:
12073:
12059:
11783:
11766:
11662:
10785:
9976:
9525:
5728:
3320:{\textstyle \left|f'(t_{i})\right|=\int _{0}^{1}\left|f'(t_{i})\right|d\theta }
315:
12499:
10087:
4597:{\displaystyle L(f)=\sup \sum _{i=1}^{N}{\bigg |}f(t_{i})-f(t_{i-1}){\bigg |}}
184:
13186:
12955:
12810:
12696:
12400:
12375:
11891:
11792:
11697:
11667:
10869:
10186:
10130:
9881:
9205:
7247:{\displaystyle \left|\left(\mathbf {x} \circ \mathbf {C} \right)'(t)\right|.}
6450:{\displaystyle \left|\left(\mathbf {x} \circ \mathbf {C} \right)'(t)\right|.}
10229:(Geometric dissertation on curved lines in comparison with straight lines).
10077:
10072:
6885:
coefficient), so the integrand of the arc length integral can be written as
12965:
12935:
12800:
12363:
11702:
9889:
9650:
9110:
192:
69:
11988:
De
Linearum Curvarum cum Lineis Rectis Comparatione Dissertatio Geometrica
10227:
De linearum curvarum cum lineis rectis comparatione dissertatio geometrica
211:
of the lengths of each linear segment; that approximation is known as the
12213:
12155:
12077:
11758:
11646:
10163:
9654:
8299:
6390:
be a curve on this surface. The integrand of the arc length integral is
5324:
11974:(2nd ed.). Amsterdam: Louis & Daniel Elzevir. pp. 517–520.
10082:
6959:{\displaystyle {\sqrt {g_{ab}\left(u^{a}\right)'\left(u^{b}\right)'\,}}}
12930:
12862:
12616:
12353:
12343:
12286:
11937:
11642:
10118:
9875:
9405:
6458:
11649:
elapsed along the world line, and arc length of a spacelike curve the
7178:{\displaystyle \mathbf {x} (r,\theta )=(r\cos \theta ,r\sin \theta ).}
13124:
12872:
12867:
12178:
12064:
11712:
10232:
9963:
8707:
So for a curve expressed in spherical coordinates, the arc length is
5732:
4815:
374:
208:
84:
11929:
13119:
12621:
12147:
11682:
11136:{\displaystyle \gamma (0)=\mathbf {x} ,\,\,\gamma (1)=\mathbf {y} }
10240:
Building on his previous work with tangents, Fermat used the curve
10222:
10175:
10126:
9181:
5716:
5700:
10819:
10121:
had pioneered a way of finding the area beneath a curve with his "
9888:(or kilometre), were originally defined so the lengths of arcs of
7610:
So for a curve expressed in polar coordinates, the arc length is:
5418:, then it is simply a special case of a parametric equation where
12970:
12223:
10452:
a relatively good approximation for the length of the curve from
10286:
10167:
10134:
10023:
6274:. This means it is possible to evaluate this integral to almost
5708:
5315:
1577:. In the below step, the following equivalent expression is used.
11984:
11954:
Tractatus Duo. Prior, De
Cycloide et de Corporibus inde Genitis…
11767:"Arc length as a global conformal parameter for analytic curves"
4693:{\displaystyle a=t_{0}<t_{1}<\dots <t_{N-1}<t_{N}=b}
405:
of the sum of linear segment lengths for a regular partition of
144:, then the curve is rectifiable (i.e., it has a finite length).
13139:
12203:
10727:
9466:
9157:
9153:
5704:
4306:
in this limit, and the right side of this equality is just the
1570:{\displaystyle dt=(t_{i}-t_{i-1})\,d\theta =\Delta t\,d\theta }
200:
5727:. The lack of a closed form solution for the arc length of an
12218:
10719:
10298:
9885:
176:
167:
Approximation to a curve by multiple linear segments, called
62:
30:
12113:
Illustrates numerical solution of finding length of a curve.
9782:(one turn is a complete rotation, or 360°, or 400 grads, or
9701:{\displaystyle s={\frac {\pi r\theta }{200{\text{ grad}}}},}
6025:{\displaystyle 1+(dy/dx)^{2}=1{\big /}\left(1-x^{2}\right),}
163:
12116:
4886:{\displaystyle g=f\circ \varphi ^{-1}:\to \mathbb {R} ^{n}}
10695:{\displaystyle AC=\varepsilon {\sqrt {1+{9 \over 4}a\,}}.}
8003:{\displaystyle \mathbf {C} (t)=(r(t),\theta (t),\phi (t))}
5486:
of each infinitesimal segment of the arc can be given by:
4758:
A curve can be parameterized in infinitely many ways. Let
437:
as the number of segments approaches infinity. This means
294:
or "direction" with respect to a reference point taken as
91:
function (i.e., the derivative is a continuous function)
10426:{\displaystyle y={3 \over 2}a^{\frac {1}{2}}(x-a)+f(a).}
10189:'s discovery of the first rectification of a nontrivial
9749:{\displaystyle s={\frac {C\theta }{400{\text{ grad}}}}.}
61:
is the distance between two points along a section of a
10804:) for any open set with 0 as one of its delimiters and
9572:{\displaystyle s={\frac {\pi r\theta }{180^{\circ }}},}
9378:
8385:
differ are zero, so the squared norm of this vector is
7342:
So the squared integrand of the arc length integral is
3938:{\displaystyle \varepsilon N\Delta t=\varepsilon (b-a)}
720:{\displaystyle \Delta t={\frac {b-a}{N}}=t_{i}-t_{i-1}}
11764:
8338:{\displaystyle \mathbf {x} _{i}\cdot \mathbf {x} _{j}}
5873:
3484:
3333:
3224:
11616:
11596:
11534:
11310:
11287:
11177:
11154:
11087:
10967:
10942:
10922:
10878:
10854:
10830:
10737:
10652:
10477:
10358:
10310:
10249:
10138:
circle, they were able to find approximate values of
10008:
9988:
9948:
9928:
9898:
9811:
9788:
9764:
9714:
9663:
9635:
9585:
9534:
9510:
9475:
9451:
9414:
9384:
9351:
9316:
9293:
9258:
9234:
9214:
9190:
9166:
9138:
8922:
8713:
8391:
8371:
8351:
8307:
8193:
8076:
8056:
8036:
8016:
7934:
7905:
7870:
7616:
7350:
7260:
7193:
7115:
7046:
7005:
6972:
6891:
6857:
6629:
6469:
6396:
6329:
6292:
6157:
6040:
5943:
5804:
5758:
5606:
5494:
5450:
5424:
5400:
5362:
5333:
5304:
4924:
4899:
4824:
4764:
4741:
4706:
4616:
4493:
4452:
4399:
4360:
4316:
4135:
4115:
4095:
4069:
4034:
4006:
3951:
3901:
3792:
3579:
3539:
2605:
2385:
2200:
2172:
2134:
2079:
1928:
1890:
1870:
1841:
1797:
1764:
1583:
1499:
1447:
1409:
1329:
1097:
782:
733:
657:
577:
445:
411:
387:
326:
269:
241:
97:
11765:
Nestoridis, Vassili; Papadopoulos, Athanase (2017).
8010:
be a curve expressed in spherical coordinates where
1084:
The last equality is proved by the following steps:
12054:
9620:{\displaystyle s={\frac {C\theta }{360^{\circ }}}.}
11733:
11625:
11602:
11579:
11513:
11293:
11269:
11160:
11135:
11063:
10948:
10928:
10908:
10860:
10836:
10769:
10694:
10632:
10425:
10338:
10274:
10014:
9994:
9954:
9934:
9910:
9839:
9797:
9770:
9748:
9700:
9641:
9619:
9571:
9516:
9493:
9457:
9432:
9393:
9369:
9337:
9299:
9279:
9240:
9220:
9196:
9172:
9144:
9089:
8906:
8697:
8377:
8357:
8337:
8290:
8177:
8062:
8042:
8022:
8002:
7917:
7891:
7854:
7600:
7334:
7246:
7177:
7099:
7024:
6991:
6958:
6873:
6841:
6611:
6449:
6382:
6315:
6228:
6119:
6024:
5929:
5859:
5790:
5685:
5588:
5474:
5436:
5406:
5386:
5348:
5294:
4908:
4885:
4806:
4747:
4727:
4692:
4596:
4470:
4438:
4381:
4346:
4298:
4121:
4101:
4081:
4055:
4021:
3992:
3937:
3887:
3776:
3563:
3525:
3470:
3319:
3208:
2587:
2366:
2186:
2158:
2120:
2065:
1914:
1876:
1856:
1815:
1783:
1748:
1569:
1485:
1433:
1395:
1315:
1074:
766:
719:
643:
561:
429:
393:
365:
275:
247:
136:
12088:Calculus Study Guide – Arc Length (Rectification)
11771:Journal of Mathematical Analysis and Applications
7100:{\displaystyle \mathbf {C} (t)=(r(t),\theta (t))}
6554:
6531:
6170:
5243:
5219:
5147:
5114:
5073:
5023:
4989:
4965:
4589:
4535:
3564:{\displaystyle \Delta t<\delta (\varepsilon )}
1915:{\displaystyle \Delta t<\delta (\varepsilon )}
1055:
1031:
891:
837:
554:
500:
13184:
4509:
3993:{\displaystyle N>(b-a)/\delta (\varepsilon )}
1791:is a continuous function from a closed interval
1396:{\displaystyle t=t_{i-1}+\theta (t_{i}-t_{i-1})}
900:
799:
462:
12103:by Chad Pierson, Josh Fritz, and Angela Sharp,
10820:Generalization to (pseudo-)Riemannian manifolds
9125:, since the Latin word for length (or size) is
5867:corresponds to a quarter of the circle. Since
199:. Since it is straightforward to calculate the
11580:{\displaystyle \gamma '(t)\in T_{\gamma (t)}M}
10816:are used to quantify the size of such curves.
10708:
8030:is the polar angle measured from the positive
6032:the length of a quarter of the unit circle is
5930:{\textstyle dy/dx=-x{\big /}{\sqrt {1-x^{2}}}}
12132:
11731:
7187:The integrand of the arc length integral is
3526:{\textstyle N>(b-a)/\delta (\varepsilon )}
309:
11736:The Theory of Splines and Their Applications
10349:so the tangent line would have the equation
7254:The chain rule for vector fields shows that
644:{\displaystyle t_{i}=a+i(b-a)/N=a+i\Delta t}
366:{\displaystyle f\colon \to \mathbb {R} ^{n}}
235:For some curves, there is a smallest number
137:{\displaystyle f\colon \to \mathbb {R} ^{n}}
12096:The MacTutor History of Mathematics archive
11967:
11957:. Oxford: University Press. pp. 91–96.
10339:{\displaystyle {3 \over 2}a^{\frac {1}{2}}}
9653:(100 grads, or grades, or gradians are one
9248:is the angle which the arc subtends at the
9228:is the length of an arc of the circle, and
7864:The second expression is for a polar graph
7035:
6383:{\displaystyle \mathbf {C} (t)=(u(t),v(t))}
12139:
12125:
12010:. Vanderbilt Univ. Press. pp. 63–90.
11916:(February 1953). "The Lengths of Curves".
9840:{\displaystyle s=C\theta /1{\text{ turn}}}
4755:is merely continuous, not differentiable.
4056:{\displaystyle \delta (\varepsilon )\to 0}
1089:The second fundamental theorem of calculus
13167:Regiomontanus' angle maximization problem
12008:Curve and Surface Design: Saint-Malo 1999
11985:M.P.E.A.S. (pseudonym of Fermat) (1660).
11782:
11112:
11111:
10686:
10271:
10236:Fermat's method of determining arc length
9853:
9075:
8892:
7840:
7731:
6953:
6113:
5671:
5574:
5336:
4873:
4426:
1560:
1544:
353:
124:
13010:
11912:
11869:
10726:
10718:
10231:
9918:applies in the following circumstances:
5742:
5314:
2597:Terms are rearranged so that it becomes
162:
37:
29:
12515:Differentiating under the integral sign
12005:
11064:{\displaystyle \gamma (t)=,\quad t\in }
6457:Evaluating the derivative requires the
2377:With the above step result, it becomes
1823:to the set of real numbers, thus it is
151:led to a general formula that provides
14:
13185:
11950:
11838:
8187:Using the chain rule again shows that
4610:is taken over all possible partitions
4439:{\displaystyle f:\to \mathbb {R} ^{n}}
2121:{\displaystyle \Delta t=t_{i}-t_{i-1}}
83:If a curve can be parameterized as an
12391:Inverse functions and differentiation
12120:
12055:
11858:CRC Handbook of Chemistry and Physics
11805:
10909:{\displaystyle \gamma :\rightarrow M}
10103:
10061:
6281:
6266:differs from the true length by only
1857:{\displaystyle \delta (\varepsilon )}
10460:. To find the length of the segment
10275:{\displaystyle y=x^{\frac {3}{2}}\,}
10068:Archimedean spiral § Arc length
6278:with only 16 integrand evaluations.
5791:{\displaystyle y={\sqrt {1-x^{2}}}.}
2187:{\displaystyle {\ce {N\to \infty }}}
290:can be defined to convey a sense of
183:can be approximated by connecting a
11810:Principles of Mathematical Analysis
9501:This is a definition of the radian.
6621:The squared norm of this vector is
6133:rule estimate for this integral of
4814:be any continuously differentiable
158:
24:
13208:One-dimensional coordinate systems
12189:Free variables and bound variables
12105:The Wolfram Demonstrations Project
12082:The Wolfram Demonstrations Project
9104:
6535:
5735:arc led to the development of the
5305:Finding arc lengths by integration
4347:{\displaystyle \left|f'(t)\right|}
4290:
4226:
4213:
4013:
3908:
3768:
3580:
3540:
3033:
2822:
2611:
2576:
2512:
2355:
2291:
2278:
2180:
2080:
1891:
1831:, so there is a positive real and
1636:
1554:
1212:
1005:
992:
910:
809:
658:
635:
472:
203:of each linear segment (using the
25:
13219:
12994:The Method of Mechanical Theorems
12025:
11918:The American Mathematical Monthly
11641:, arc length of timelike curves (
10093:Sine and cosine § Arc length
9377:This equation is a definition of
9307:are expressed in the same units.
6316:{\displaystyle \mathbf {x} (u,v)}
5598:The arc length is then given by:
5260:using integration by substitution
4082:{\displaystyle \varepsilon \to 0}
12549:Partial fractions in integration
12465:Stochastic differential equation
11129:
11104:
10770:{\displaystyle x\cdot \sin(1/x)}
10204:
8540:
8525:
8477:
8462:
8414:
8399:
8325:
8310:
8267:
8244:
8221:
8209:
8201:
8078:
7936:
7495:
7480:
7439:
7424:
7377:
7373:
7362:
7358:
7311:
7288:
7276:
7268:
7214:
7206:
7117:
7048:
6714:
6691:
6660:
6637:
6588:
6565:
6515:
6500:
6485:
6477:
6417:
6409:
6331:
6294:
6149:differs from the true length of
5349:{\displaystyle \mathbb {R} ^{2}}
767:{\displaystyle i=0,1,\dotsc ,N.}
12687:Jacobian matrix and determinant
12542:Tangent half-angle substitution
12510:Fundamental theorem of calculus
11978:
11884:10.4169/college.math.j.46.3.162
11872:The College Mathematics Journal
11281:or, choosing local coordinates
11039:
10145:
10098:Triangle wave § Arc length
9442:For an arbitrary circular arc:
9099:
5311:Differential geometry of curves
5257:
5178:
3029:
2818:
1784:{\displaystyle \left|f'\right|}
50:as a function of its parameter
12763:Arithmetico-geometric sequence
12455:Ordinary differential equation
11961:
11944:
11906:
11863:
11851:
11832:
11799:
11725:
11569:
11563:
11549:
11543:
11489:
11486:
11480:
11474:
11444:
11441:
11435:
11429:
11410:
11407:
11404:
11398:
11392:
11386:
11320:
11314:
11256:
11250:
11242:
11236:
11232:
11226:
11214:
11209:
11187:
11181:
11122:
11116:
11097:
11091:
11058:
11046:
11033:
11030:
11024:
11002:
10996:
10983:
10977:
10971:
10900:
10897:
10885:
10764:
10750:
10417:
10411:
10402:
10390:
8213:
8197:
8169:
8106:
8100:
8082:
7997:
7994:
7988:
7979:
7973:
7964:
7958:
7952:
7946:
7940:
7886:
7880:
7785:
7772:
7764:
7751:
7280:
7264:
7233:
7227:
7169:
7139:
7133:
7121:
7094:
7091:
7085:
7076:
7070:
7064:
7058:
7052:
6732:
6686:
6525:
6495:
6489:
6473:
6436:
6430:
6377:
6374:
6368:
6359:
6353:
6347:
6341:
6335:
6310:
6298:
5971:
5950:
5466:
5460:
5378:
5372:
5282:
5276:
5238:
5232:
5166:
5160:
5142:
5139:
5133:
5127:
5068:
5062:
5051:
5048:
5042:
5036:
4984:
4978:
4938:
4932:
4868:
4865:
4853:
4801:
4789:
4786:
4783:
4771:
4719:
4707:
4584:
4565:
4556:
4543:
4503:
4497:
4465:
4453:
4421:
4418:
4406:
4373:
4361:
4336:
4330:
4282:
4269:
4208:
4189:
4180:
4167:
4073:
4047:
4044:
4038:
4010:
3987:
3981:
3970:
3958:
3932:
3920:
3871:
3858:
3819:
3806:
3746:
3733:
3700:
3697:
3665:
3640:
3558:
3552:
3520:
3514:
3503:
3491:
3449:
3436:
3412:
3409:
3377:
3352:
3303:
3290:
3251:
3238:
3185:
3172:
3148:
3145:
3113:
3088:
3003:
2990:
2942:
2939:
2907:
2882:
2792:
2779:
2731:
2728:
2696:
2671:
2568:
2555:
2495:
2492:
2460:
2435:
2347:
2334:
2273:
2254:
2245:
2232:
2177:
2153:
2141:
2044:
2031:
2007:
2004:
1972:
1947:
1909:
1903:
1851:
1845:
1810:
1798:
1731:
1728:
1696:
1671:
1631:
1612:
1603:
1590:
1541:
1509:
1480:
1448:
1428:
1416:
1390:
1358:
1301:
1298:
1266:
1241:
1197:
1191:
1142:
1123:
1114:
1101:
1050:
1044:
987:
968:
959:
946:
907:
886:
867:
858:
845:
806:
792:
786:
612:
600:
549:
530:
521:
508:
469:
455:
449:
424:
412:
348:
345:
333:
316:Curve § Length of a curve
191:on the curve using (straight)
119:
116:
104:
13:
1:
12586:Integro-differential equation
12460:Partial differential equation
11814:. McGraw-Hill, Inc. pp.
11777:(2). Elsevier BV: 1505–1515.
11718:
9252:of the circle. The distances
6323:be a surface mapping and let
4807:{\displaystyle \varphi :\to }
4022:{\displaystyle N\to \infty ,}
12146:
12111:Length of a Curve Experiment
11839:Suplee, Curt (2 July 2009).
10846:(pseudo-)Riemannian manifold
10108:
7892:{\displaystyle r=r(\theta )}
1877:{\displaystyle \varepsilon }
1833:monotonically non-decreasing
7:
12740:Generalized Stokes' theorem
12527:Integration by substitution
12038:Encyclopedia of Mathematics
11991:. Toulouse: Arnaud Colomer.
11971:Renati Des-Cartes Geometria
11656:
10709:Curves with infinite length
10643:which, when solved, yields
10133:. People began to inscribe
9494:{\displaystyle s=r\theta .}
9121:Arc lengths are denoted by
9117:Circumference § Circle
5699:for arc length include the
5416:continuously differentiable
5356:is defined by the equation
4391:continuously differentiable
3218:where in the leftmost side
2166:. Let's consider the limit
2159:{\displaystyle \theta \in }
1434:{\displaystyle \theta \in }
379:continuously differentiable
89:continuously differentiable
10:
13224:
12269:(ε, δ)-definition of limit
11999:
11784:10.1016/j.jmaa.2016.02.031
11740:. Academic Press. p.
10712:
10162:in 1645 (some sources say
10088:Parabola § Arc length
9942:is in nautical miles, and
9869:
9863:
9857:
9114:
9108:
5860:{\displaystyle x\in \left}
5308:
313:
310:Formula for a smooth curve
13162:Proof that 22/7 exceeds π
13099:
13077:
13003:
12951:Gottfried Wilhelm Leibniz
12921:
12898:e (mathematical constant)
12883:
12755:
12662:
12594:
12475:
12277:
12232:
12154:
11841:"Special Publication 811"
11590:is the tangent vector of
10185:In 1659, Wallis credited
10078:Ellipse § Arc length
10073:Cycloid § Arc length
9911:{\displaystyle s=\theta }
9880:Two units of length, the
9866:Geodesics on an ellipsoid
9338:{\displaystyle C=2\pi r,}
7918:{\displaystyle t=\theta }
2194:of the following formula,
1864:of positive real numbers
263:is defined as the number
12913:Stirling's approximation
12386:Implicit differentiation
12334:Rules of differentiation
12101:Arc Length Approximation
12050:The History of Curvature
11732:Ahlberg; Nilson (1967).
9433:{\displaystyle s=\pi r.}
9370:{\displaystyle C=\pi d.}
9132:In the following lines,
7036:Other coordinate systems
4478:is always finite, i.e.,
298:in the curve (see also:
13147:Euler–Maclaurin formula
13052:trigonometric functions
12505:Constant of integration
11693:Integral approximations
11603:{\displaystyle \gamma }
11161:{\displaystyle \gamma }
10808:(0) = 0. Sometimes the
10083:Helix § Arc length
10015:{\displaystyle \theta }
9955:{\displaystyle \theta }
9771:{\displaystyle \theta }
9642:{\displaystyle \theta }
9517:{\displaystyle \theta }
9458:{\displaystyle \theta }
9241:{\displaystyle \theta }
8023:{\displaystyle \theta }
7025:{\displaystyle u^{2}=v}
6992:{\displaystyle u^{1}=u}
5475:{\displaystyle y=f(t).}
5387:{\displaystyle y=f(x),}
5189: is non-decreasing
13116:Differential geometry
12961:Infinitesimal calculus
12664:Multivariable calculus
12612:Directional derivative
12418:Second derivative test
12396:Logarithmic derivative
12369:General Leibniz's rule
12264:Order of approximation
11806:Rudin, Walter (1976).
11708:Multivariable calculus
11627:
11604:
11581:
11515:
11372:
11340:
11295:
11271:
11207:
11162:
11137:
11065:
10950:
10930:
10910:
10862:
10838:
10777:
10771:
10724:
10696:
10634:
10427:
10340:
10276:
10237:
10160:Evangelista Torricelli
10115:history of mathematics
10016:
10002:is in kilometres, and
9996:
9956:
9936:
9912:
9854:Great circles on Earth
9841:
9799:
9772:
9750:
9702:
9643:
9621:
9573:
9518:
9495:
9459:
9434:
9395:
9371:
9339:
9301:
9281:
9280:{\displaystyle r,d,C,}
9242:
9222:
9198:
9174:
9146:
9091:
8908:
8699:
8379:
8359:
8339:
8292:
8179:
8064:
8044:
8024:
8004:
7919:
7893:
7856:
7602:
7336:
7248:
7179:
7101:
7026:
6993:
6960:
6883:first fundamental form
6875:
6874:{\displaystyle g_{ij}}
6843:
6613:
6451:
6384:
6317:
6230:
6121:
6026:
5931:
5861:
5792:
5687:
5590:
5476:
5438:
5408:
5388:
5350:
5320:
5296:
4910:
4887:
4808:
4749:
4729:
4694:
4598:
4532:
4472:
4440:
4383:
4348:
4300:
4255:
4156:
4123:
4103:
4089:thus the left side of
4083:
4057:
4023:
3994:
3939:
3889:
3778:
3606:
3565:
3527:
3472:
3321:
3210:
3059:
2848:
2637:
2589:
2541:
2406:
2368:
2320:
2221:
2188:
2160:
2122:
2067:
1916:
1878:
1858:
1817:
1785:
1750:
1571:
1487:
1435:
1397:
1317:
1076:
935:
834:
768:
721:
645:
563:
497:
431:
401:can be defined as the
395:
367:
277:
249:
172:
149:infinitesimal calculus
138:
55:
35:
27:Distance along a curve
13035:logarithmic functions
13030:exponential functions
12946:Generality of algebra
12824:Tests of convergence
12450:Differential equation
12434:Further applications
12423:Extreme value theorem
12413:First derivative test
12307:Differential operator
12279:Differential calculus
11951:Wallis, John (1659).
11628:
11605:
11582:
11516:
11346:
11326:
11296:
11272:
11193:
11163:
11138:
11066:
10956:parametric equations
10951:
10931:
10911:
10863:
10839:
10772:
10730:
10722:
10697:
10635:
10440:by a small amount to
10428:
10341:
10277:
10235:
10152:transcendental curves
10017:
9997:
9957:
9937:
9913:
9864:Further information:
9860:Great-circle distance
9842:
9800:
9798:{\displaystyle 2\pi }
9773:
9751:
9708:which is the same as
9703:
9644:
9622:
9579:which is the same as
9574:
9519:
9496:
9460:
9435:
9396:
9394:{\displaystyle \pi .}
9372:
9345:which is the same as
9340:
9302:
9282:
9243:
9223:
9199:
9175:
9147:
9115:Further information:
9092:
8909:
8700:
8380:
8360:
8340:
8293:
8180:
8065:
8063:{\displaystyle \phi }
8045:
8025:
8005:
7920:
7894:
7857:
7603:
7337:
7249:
7180:
7102:
7027:
6994:
6961:
6876:
6844:
6614:
6452:
6385:
6318:
6231:
6122:
6027:
5932:
5862:
5793:
5749:numerical integration
5743:Numerical integration
5697:closed-form solutions
5688:
5591:
5477:
5439:
5409:
5389:
5351:
5318:
5297:
4911:
4888:
4809:
4750:
4730:
4695:
4599:
4512:
4473:
4441:
4384:
4349:
4301:
4235:
4136:
4124:
4104:
4084:
4058:
4024:
3995:
3940:
3890:
3779:
3586:
3566:
3528:
3473:
3322:
3211:
3039:
2828:
2617:
2590:
2521:
2386:
2369:
2300:
2201:
2189:
2161:
2123:
2068:
1917:
1879:
1859:
1818:
1786:
1751:
1572:
1488:
1436:
1398:
1318:
1077:
915:
814:
769:
722:
646:
564:
477:
432:
396:
368:
278:
250:
166:
153:closed-form solutions
139:
41:
33:
13100:Miscellaneous topics
13040:hyperbolic functions
13025:irrational functions
12903:Exponential function
12756:Sequences and series
12522:Integration by parts
11639:theory of relativity
11614:
11594:
11532:
11308:
11285:
11175:
11152:
11085:
10965:
10940:
10920:
10876:
10852:
10828:
10735:
10650:
10475:
10356:
10308:
10247:
10195:semicubical parabola
10123:method of exhaustion
10006:
9986:
9946:
9926:
9896:
9876:Gradian § Metre
9809:
9786:
9762:
9712:
9661:
9633:
9583:
9532:
9508:
9473:
9449:
9412:
9382:
9349:
9314:
9291:
9256:
9232:
9212:
9188:
9164:
9136:
8920:
8711:
8389:
8369:
8349:
8305:
8191:
8074:
8054:
8034:
8014:
7932:
7903:
7868:
7614:
7348:
7258:
7191:
7113:
7044:
7003:
6970:
6889:
6855:
6627:
6467:
6394:
6327:
6290:
6155:
6038:
5941:
5871:
5802:
5756:
5721:semicubical parabola
5604:
5492:
5448:
5422:
5398:
5360:
5331:
4922:
4897:
4822:
4762:
4739:
4704:
4614:
4491:
4450:
4397:
4358:
4314:
4133:
4113:
4102:{\displaystyle <}
4093:
4067:
4032:
4004:
3949:
3899:
3790:
3577:
3537:
3482:
3331:
3222:
2603:
2383:
2198:
2170:
2132:
2077:
1926:
1888:
1868:
1839:
1829:Heine–Cantor theorem
1825:uniformly continuous
1795:
1762:
1581:
1497:
1445:
1407:
1327:
1095:
780:
731:
655:
575:
443:
409:
385:
324:
267:
239:
95:
13087:List of derivatives
12923:History of calculus
12838:Cauchy condensation
12735:Exterior derivative
12692:Lagrange multiplier
12428:Maximum and minimum
12259:Limit of a sequence
12247:Limit of a function
12194:Graph of a function
12174:Continuous function
12093:Famous Curves Index
12033:"Rectifiable curve"
11168:, is defined to be
10810:Hausdorff dimension
10466:Pythagorean theorem
10436:Next, he increased
10211:Hendrik van Heuraet
10166:in the 1650s), the
8951:
8742:
7789:
7645:
6461:for vector fields:
6248:Gaussian quadrature
6212:
6082:
5627:
5437:{\displaystyle x=t}
5216:
5111:
5020:
4962:
3844:
3631:
3276:
3074:
2976:
2868:
2765:
2662:
2426:
1662:
1232:
1182:
1028:
205:Pythagorean theorem
74:curve rectification
13020:rational functions
12987:Method of Fluxions
12833:Alternating series
12730:Differential forms
12712:Partial derivative
12672:Divergence theorem
12554:Quadratic integral
12322:Leibniz's notation
12312:Mean value theorem
12297:Partial derivative
12242:Indeterminate form
12057:Weisstein, Eric W.
11688:Intrinsic equation
11626:{\displaystyle t.}
11623:
11600:
11577:
11511:
11291:
11267:
11158:
11133:
11061:
10946:
10926:
10906:
10858:
10834:
10778:
10767:
10725:
10692:
10630:
10628:
10423:
10336:
10272:
10238:
10156:logarithmic spiral
10104:Historical methods
10062:Other simple cases
10012:
9992:
9952:
9932:
9908:
9872:Length of a degree
9837:
9795:
9768:
9746:
9698:
9639:
9617:
9569:
9514:
9491:
9455:
9430:
9391:
9367:
9335:
9297:
9277:
9238:
9218:
9194:
9170:
9142:
9087:
8923:
8904:
8714:
8695:
8375:
8355:
8335:
8288:
8175:
8060:
8040:
8020:
8000:
7915:
7889:
7852:
7743:
7617:
7598:
7332:
7244:
7175:
7097:
7022:
6989:
6956:
6871:
6839:
6609:
6447:
6380:
6313:
6282:Curve on a surface
6226:
6167:
6117:
6041:
6022:
5927:
5857:
5788:
5737:elliptic integrals
5713:logarithmic spiral
5683:
5613:
5586:
5484:Euclidean distance
5472:
5434:
5404:
5384:
5346:
5321:
5292:
5290:
5202:
5097:
5006:
4948:
4909:{\displaystyle f.}
4906:
4883:
4804:
4745:
4725:
4690:
4594:
4468:
4436:
4379:
4344:
4296:
4129:. In other words,
4119:
4099:
4079:
4053:
4019:
3990:
3935:
3885:
3830:
3774:
3617:
3561:
3523:
3468:
3317:
3262:
3206:
3204:
3060:
2962:
2854:
2751:
2648:
2585:
2412:
2364:
2184:
2156:
2118:
2063:
1912:
1874:
1854:
1813:
1781:
1746:
1648:
1567:
1483:
1431:
1393:
1313:
1218:
1148:
1072:
1014:
914:
813:
764:
717:
641:
559:
476:
427:
391:
363:
273:
245:
173:
134:
56:
48:logarithmic spiral
36:
13193:Integral calculus
13180:
13179:
13106:Complex calculus
13095:
13094:
12976:Law of Continuity
12908:Natural logarithm
12893:Bernoulli numbers
12884:Special functions
12843:Direct comparison
12707:Multiple integral
12581:Integral equation
12477:Integral calculus
12408:Stationary points
12382:Other techniques
12327:Newton's notation
12292:Second derivative
12184:Finite difference
12017:978-0-8265-1356-4
11825:978-0-07-054235-8
11678:Elliptic integral
11653:along the curve.
11503:
11501:
11456:
11294:{\displaystyle x}
10949:{\displaystyle n}
10929:{\displaystyle M}
10861:{\displaystyle g}
10837:{\displaystyle M}
10814:Hausdorff measure
10782:arbitrarily large
10715:Coastline paradox
10687:
10681:
10616:
10562:
10448:, making segment
10387:
10373:
10333:
10319:
10268:
10221:the area under a
10180:Gottfried Leibniz
10174:in 1658, and the
9995:{\displaystyle s}
9935:{\displaystyle s}
9835:
9741:
9738:
9693:
9690:
9612:
9564:
9300:{\displaystyle s}
9221:{\displaystyle s}
9197:{\displaystyle C}
9173:{\displaystyle d}
9145:{\displaystyle r}
9076:
9063:
9025:
8977:
8893:
8880:
8816:
8768:
8378:{\displaystyle j}
8358:{\displaystyle i}
8043:{\displaystyle z}
7899:parameterized by
7841:
7815:
7732:
7719:
7671:
6954:
6552:
6512:
6276:machine precision
6250:rule estimate of
6246:and the 16-point
6224:
6201:
6184:
6111:
6110:
6071:
6054:
5925:
5842:
5824:
5783:
5672:
5659:
5575:
5562:
5526:
5407:{\displaystyle f}
5261:
5250:
5190:
5182:
5181:in the case
5171:
5080:
4996:
4748:{\displaystyle f}
4728:{\displaystyle .}
4382:{\displaystyle .}
4220:
4122:{\displaystyle 0}
3705:
3195:
2952:
2736:
2500:
2285:
2176:
1827:according to the
1736:
1643:
1306:
1202:
1062:
999:
899:
798:
683:
461:
394:{\displaystyle f}
300:curve orientation
288:signed arc length
276:{\displaystyle L}
248:{\displaystyle L}
230:arbitrarily small
78:rectifiable curve
18:Rectifiable curve
16:(Redirected from
13215:
13110:Contour integral
13008:
13007:
12858:Limit comparison
12767:Types of series
12726:Advanced topics
12717:Surface integral
12561:Trapezoidal rule
12500:Basic properties
12495:Riemann integral
12443:Taylor's theorem
12169:Concave function
12164:Binomial theorem
12141:
12134:
12127:
12118:
12117:
12070:
12069:
12046:
12021:
11993:
11992:
11982:
11976:
11975:
11965:
11959:
11958:
11948:
11942:
11941:
11910:
11904:
11903:
11867:
11861:
11855:
11849:
11848:
11836:
11830:
11829:
11813:
11803:
11797:
11796:
11786:
11762:
11756:
11755:
11739:
11729:
11632:
11630:
11629:
11624:
11609:
11607:
11606:
11601:
11586:
11584:
11583:
11578:
11573:
11572:
11542:
11520:
11518:
11517:
11512:
11504:
11502:
11500:
11492:
11473:
11472:
11459:
11457:
11455:
11447:
11428:
11427:
11414:
11385:
11384:
11371:
11366:
11342:
11339:
11334:
11300:
11298:
11297:
11292:
11276:
11274:
11273:
11268:
11260:
11259:
11245:
11239:
11225:
11217:
11212:
11206:
11201:
11167:
11165:
11164:
11159:
11142:
11140:
11139:
11134:
11132:
11107:
11070:
11068:
11067:
11062:
11023:
11022:
10995:
10994:
10955:
10953:
10952:
10947:
10935:
10933:
10932:
10927:
10915:
10913:
10912:
10907:
10867:
10865:
10864:
10859:
10843:
10841:
10840:
10835:
10776:
10774:
10773:
10768:
10760:
10701:
10699:
10698:
10693:
10688:
10682:
10674:
10666:
10639:
10637:
10636:
10631:
10629:
10625:
10621:
10617:
10609:
10596:
10595:
10580:
10576:
10575:
10563:
10555:
10550:
10549:
10534:
10530:
10529:
10514:
10513:
10494:
10493:
10432:
10430:
10429:
10424:
10389:
10388:
10380:
10374:
10366:
10345:
10343:
10342:
10337:
10335:
10334:
10326:
10320:
10312:
10281:
10279:
10278:
10273:
10270:
10269:
10261:
10215:Pierre de Fermat
10199:Gulielmus Nelius
10172:Christopher Wren
10113:For much of the
10057:
10056:
10053:
10043:
10042:
10036:
10035:
10021:
10019:
10018:
10013:
10001:
9999:
9998:
9993:
9975:
9974:
9970:
9961:
9959:
9958:
9953:
9941:
9939:
9938:
9933:
9917:
9915:
9914:
9909:
9846:
9844:
9843:
9838:
9836:
9833:
9828:
9804:
9802:
9801:
9796:
9777:
9775:
9774:
9769:
9755:
9753:
9752:
9747:
9742:
9740:
9739:
9736:
9730:
9722:
9707:
9705:
9704:
9699:
9694:
9692:
9691:
9688:
9682:
9671:
9648:
9646:
9645:
9640:
9626:
9624:
9623:
9618:
9613:
9611:
9610:
9601:
9593:
9578:
9576:
9575:
9570:
9565:
9563:
9562:
9553:
9542:
9523:
9521:
9520:
9515:
9500:
9498:
9497:
9492:
9464:
9462:
9461:
9456:
9439:
9437:
9436:
9431:
9404:If the arc is a
9400:
9398:
9397:
9392:
9376:
9374:
9373:
9368:
9344:
9342:
9341:
9336:
9306:
9304:
9303:
9298:
9286:
9284:
9283:
9278:
9247:
9245:
9244:
9239:
9227:
9225:
9224:
9219:
9203:
9201:
9200:
9195:
9179:
9177:
9176:
9171:
9151:
9149:
9148:
9143:
9105:Arcs of circles
9096:
9094:
9093:
9088:
9077:
9074:
9073:
9068:
9064:
9062:
9054:
9046:
9036:
9035:
9030:
9026:
9024:
9016:
9008:
9001:
9000:
8988:
8987:
8982:
8978:
8976:
8968:
8960:
8953:
8950:
8949:
8948:
8938:
8937:
8936:
8913:
8911:
8910:
8905:
8894:
8891:
8890:
8885:
8881:
8879:
8871:
8863:
8850:
8849:
8840:
8839:
8827:
8826:
8821:
8817:
8815:
8807:
8799:
8792:
8791:
8779:
8778:
8773:
8769:
8767:
8759:
8751:
8744:
8741:
8740:
8739:
8729:
8728:
8727:
8704:
8702:
8701:
8696:
8691:
8690:
8685:
8681:
8662:
8661:
8652:
8651:
8639:
8638:
8633:
8629:
8616:
8615:
8603:
8602:
8597:
8593:
8577:
8576:
8571:
8567:
8554:
8550:
8549:
8548:
8543:
8534:
8533:
8528:
8514:
8513:
8508:
8504:
8491:
8487:
8486:
8485:
8480:
8471:
8470:
8465:
8451:
8447:
8446:
8445:
8428:
8424:
8423:
8422:
8417:
8408:
8407:
8402:
8384:
8382:
8381:
8376:
8364:
8362:
8361:
8356:
8344:
8342:
8341:
8336:
8334:
8333:
8328:
8319:
8318:
8313:
8297:
8295:
8294:
8289:
8284:
8276:
8275:
8270:
8261:
8253:
8252:
8247:
8238:
8230:
8229:
8224:
8212:
8204:
8184:
8182:
8181:
8176:
8081:
8069:
8067:
8066:
8061:
8049:
8047:
8046:
8041:
8029:
8027:
8026:
8021:
8009:
8007:
8006:
8001:
7939:
7924:
7922:
7921:
7916:
7898:
7896:
7895:
7890:
7861:
7859:
7858:
7853:
7842:
7839:
7838:
7826:
7825:
7820:
7816:
7814:
7806:
7798:
7791:
7788:
7784:
7783:
7767:
7763:
7762:
7733:
7730:
7729:
7724:
7720:
7718:
7710:
7702:
7695:
7694:
7682:
7681:
7676:
7672:
7670:
7662:
7654:
7647:
7644:
7643:
7642:
7632:
7631:
7630:
7607:
7605:
7604:
7599:
7594:
7593:
7588:
7584:
7571:
7570:
7558:
7557:
7552:
7548:
7532:
7531:
7526:
7522:
7509:
7505:
7504:
7503:
7498:
7489:
7488:
7483:
7469:
7461:
7453:
7449:
7448:
7447:
7442:
7433:
7432:
7427:
7410:
7409:
7404:
7400:
7387:
7383:
7382:
7381:
7380:
7367:
7366:
7365:
7341:
7339:
7338:
7333:
7328:
7320:
7319:
7314:
7305:
7297:
7296:
7291:
7279:
7271:
7253:
7251:
7250:
7245:
7240:
7236:
7226:
7222:
7218:
7217:
7209:
7184:
7182:
7181:
7176:
7120:
7106:
7104:
7103:
7098:
7051:
7031:
7029:
7028:
7023:
7015:
7014:
6998:
6996:
6995:
6990:
6982:
6981:
6965:
6963:
6962:
6957:
6955:
6952:
6948:
6944:
6943:
6929:
6925:
6921:
6920:
6906:
6905:
6893:
6880:
6878:
6877:
6872:
6870:
6869:
6848:
6846:
6845:
6840:
6838:
6837:
6832:
6828:
6815:
6814:
6802:
6794:
6786:
6785:
6770:
6769:
6764:
6760:
6747:
6746:
6731:
6723:
6722:
6717:
6708:
6700:
6699:
6694:
6682:
6678:
6677:
6669:
6668:
6663:
6654:
6646:
6645:
6640:
6618:
6616:
6615:
6610:
6605:
6597:
6596:
6591:
6582:
6574:
6573:
6568:
6559:
6558:
6557:
6551:
6543:
6534:
6524:
6523:
6518:
6510:
6509:
6508:
6503:
6488:
6480:
6456:
6454:
6453:
6448:
6443:
6439:
6429:
6425:
6421:
6420:
6412:
6389:
6387:
6386:
6381:
6334:
6322:
6320:
6319:
6314:
6297:
6273:
6271:
6265:
6264:
6261:
6258:
6255:
6245:
6243:
6235:
6233:
6232:
6227:
6225:
6217:
6211:
6207:
6202:
6197:
6194:
6190:
6185:
6180:
6174:
6173:
6148:
6147:
6144:
6141:
6138:
6126:
6124:
6123:
6118:
6112:
6109:
6108:
6093:
6092:
6084:
6081:
6077:
6072:
6067:
6064:
6060:
6055:
6050:
6031:
6029:
6028:
6023:
6018:
6014:
6013:
6012:
5992:
5991:
5979:
5978:
5963:
5936:
5934:
5933:
5928:
5926:
5924:
5923:
5908:
5906:
5905:
5884:
5866:
5864:
5863:
5858:
5856:
5852:
5848:
5843:
5838:
5830:
5825:
5820:
5797:
5795:
5794:
5789:
5784:
5782:
5781:
5766:
5692:
5690:
5689:
5684:
5673:
5670:
5669:
5664:
5660:
5658:
5650:
5642:
5629:
5626:
5621:
5595:
5593:
5592:
5587:
5576:
5573:
5572:
5567:
5563:
5561:
5553:
5545:
5532:
5527:
5525:
5524:
5509:
5508:
5496:
5481:
5479:
5478:
5473:
5443:
5441:
5440:
5435:
5413:
5411:
5410:
5405:
5393:
5391:
5390:
5385:
5355:
5353:
5352:
5347:
5345:
5344:
5339:
5301:
5299:
5298:
5293:
5291:
5266:
5262:
5259:
5248:
5247:
5246:
5231:
5223:
5222:
5215:
5210:
5195:
5191:
5188:
5183:
5180:
5169:
5159:
5151:
5150:
5126:
5118:
5117:
5110:
5105:
5090:
5078:
5077:
5076:
5061:
5035:
5027:
5026:
5019:
5014:
4994:
4993:
4992:
4977:
4969:
4968:
4961:
4956:
4915:
4913:
4912:
4907:
4892:
4890:
4889:
4884:
4882:
4881:
4876:
4849:
4848:
4813:
4811:
4810:
4805:
4754:
4752:
4751:
4746:
4734:
4732:
4731:
4726:
4699:
4697:
4696:
4691:
4683:
4682:
4670:
4669:
4645:
4644:
4632:
4631:
4603:
4601:
4600:
4595:
4593:
4592:
4583:
4582:
4555:
4554:
4539:
4538:
4531:
4526:
4477:
4475:
4474:
4471:{\displaystyle }
4469:
4445:
4443:
4442:
4437:
4435:
4434:
4429:
4388:
4386:
4385:
4380:
4353:
4351:
4350:
4345:
4343:
4339:
4329:
4308:Riemann integral
4305:
4303:
4302:
4297:
4289:
4285:
4281:
4280:
4268:
4254:
4249:
4225:
4221:
4219:
4211:
4207:
4206:
4179:
4178:
4162:
4155:
4150:
4128:
4126:
4125:
4120:
4108:
4106:
4105:
4100:
4088:
4086:
4085:
4080:
4062:
4060:
4059:
4054:
4028:
4026:
4025:
4020:
3999:
3997:
3996:
3991:
3977:
3944:
3942:
3941:
3936:
3894:
3892:
3891:
3886:
3878:
3874:
3870:
3869:
3857:
3843:
3838:
3826:
3822:
3818:
3817:
3805:
3783:
3781:
3780:
3775:
3758:
3754:
3753:
3749:
3745:
3744:
3732:
3716:
3712:
3703:
3696:
3695:
3677:
3676:
3658:
3657:
3639:
3630:
3625:
3605:
3600:
3570:
3568:
3567:
3562:
3532:
3530:
3529:
3524:
3510:
3477:
3475:
3474:
3469:
3461:
3457:
3456:
3452:
3448:
3447:
3435:
3419:
3415:
3408:
3407:
3389:
3388:
3370:
3369:
3351:
3326:
3324:
3323:
3318:
3310:
3306:
3302:
3301:
3289:
3275:
3270:
3258:
3254:
3250:
3249:
3237:
3215:
3213:
3212:
3207:
3205:
3193:
3192:
3188:
3184:
3183:
3171:
3155:
3151:
3144:
3143:
3125:
3124:
3106:
3105:
3087:
3073:
3068:
3058:
3053:
3025:
3021:
3017:
3010:
3006:
3002:
3001:
2989:
2975:
2970:
2950:
2949:
2945:
2938:
2937:
2919:
2918:
2900:
2899:
2881:
2867:
2862:
2847:
2842:
2814:
2810:
2806:
2799:
2795:
2791:
2790:
2778:
2764:
2759:
2747:
2743:
2734:
2727:
2726:
2708:
2707:
2689:
2688:
2670:
2661:
2656:
2636:
2631:
2609:
2594:
2592:
2591:
2586:
2575:
2571:
2567:
2566:
2554:
2540:
2535:
2511:
2507:
2498:
2491:
2490:
2472:
2471:
2453:
2452:
2434:
2425:
2420:
2405:
2400:
2373:
2371:
2370:
2365:
2354:
2350:
2346:
2345:
2333:
2319:
2314:
2290:
2286:
2284:
2276:
2272:
2271:
2244:
2243:
2227:
2220:
2215:
2193:
2191:
2190:
2185:
2183:
2174:
2165:
2163:
2162:
2157:
2127:
2125:
2124:
2119:
2117:
2116:
2098:
2097:
2072:
2070:
2069:
2064:
2056:
2052:
2051:
2047:
2043:
2042:
2030:
2014:
2010:
2003:
2002:
1984:
1983:
1965:
1964:
1946:
1921:
1919:
1918:
1913:
1883:
1881:
1880:
1875:
1863:
1861:
1860:
1855:
1822:
1820:
1819:
1816:{\displaystyle }
1814:
1790:
1788:
1787:
1782:
1780:
1776:
1755:
1753:
1752:
1747:
1734:
1727:
1726:
1708:
1707:
1689:
1688:
1670:
1661:
1656:
1644:
1642:
1634:
1630:
1629:
1602:
1601:
1585:
1576:
1574:
1573:
1568:
1540:
1539:
1521:
1520:
1492:
1490:
1489:
1486:{\displaystyle }
1484:
1479:
1478:
1466:
1465:
1440:
1438:
1437:
1432:
1402:
1400:
1399:
1394:
1389:
1388:
1370:
1369:
1351:
1350:
1322:
1320:
1319:
1314:
1304:
1297:
1296:
1278:
1277:
1259:
1258:
1240:
1231:
1226:
1200:
1190:
1181:
1180:
1179:
1169:
1168:
1167:
1141:
1140:
1113:
1112:
1081:
1079:
1078:
1073:
1060:
1059:
1058:
1043:
1035:
1034:
1027:
1022:
1004:
1000:
998:
990:
986:
985:
958:
957:
941:
934:
929:
913:
895:
894:
885:
884:
857:
856:
841:
840:
833:
828:
812:
773:
771:
770:
765:
726:
724:
723:
718:
716:
715:
697:
696:
684:
679:
668:
650:
648:
647:
642:
619:
587:
586:
568:
566:
565:
560:
558:
557:
548:
547:
520:
519:
504:
503:
496:
491:
475:
436:
434:
433:
430:{\displaystyle }
428:
400:
398:
397:
392:
372:
370:
369:
364:
362:
361:
356:
282:
280:
279:
274:
254:
252:
251:
246:
159:General approach
143:
141:
140:
135:
133:
132:
127:
21:
13223:
13222:
13218:
13217:
13216:
13214:
13213:
13212:
13183:
13182:
13181:
13176:
13172:Steinmetz solid
13157:Integration Bee
13091:
13073:
12999:
12941:Colin Maclaurin
12917:
12885:
12879:
12751:
12745:Tensor calculus
12722:Volume integral
12658:
12633:Basic theorems
12596:Vector calculus
12590:
12471:
12438:Newton's method
12273:
12252:One-sided limit
12228:
12209:Rolle's theorem
12199:Linear function
12150:
12145:
12031:
12028:
12018:
12002:
11997:
11996:
11983:
11979:
11966:
11962:
11949:
11945:
11930:10.2307/2308256
11914:Coolidge, J. L.
11911:
11907:
11868:
11864:
11856:
11852:
11837:
11833:
11826:
11804:
11800:
11763:
11759:
11752:
11730:
11726:
11721:
11673:Crofton formula
11659:
11651:proper distance
11615:
11612:
11611:
11595:
11592:
11591:
11559:
11555:
11535:
11533:
11530:
11529:
11493:
11468:
11464:
11460:
11458:
11448:
11423:
11419:
11415:
11413:
11377:
11373:
11367:
11350:
11341:
11335:
11330:
11309:
11306:
11305:
11286:
11283:
11282:
11246:
11241:
11240:
11235:
11218:
11213:
11208:
11202:
11197:
11176:
11173:
11172:
11153:
11150:
11149:
11128:
11103:
11086:
11083:
11082:
11018:
11014:
10990:
10986:
10966:
10963:
10962:
10941:
10938:
10937:
10921:
10918:
10917:
10877:
10874:
10873:
10853:
10850:
10849:
10829:
10826:
10825:
10822:
10756:
10736:
10733:
10732:
10723:The Koch curve.
10717:
10711:
10673:
10665:
10651:
10648:
10647:
10627:
10626:
10608:
10601:
10597:
10591:
10587:
10578:
10577:
10571:
10567:
10554:
10545:
10541:
10532:
10531:
10525:
10521:
10509:
10505:
10495:
10489:
10485:
10478:
10476:
10473:
10472:
10379:
10375:
10365:
10357:
10354:
10353:
10325:
10321:
10311:
10309:
10306:
10305:
10260:
10256:
10248:
10245:
10244:
10207:
10191:algebraic curve
10148:
10111:
10106:
10064:
10054:
10051:
10049:
10040:
10038:
10037:kilometres, or
10033:
10031:
10007:
10004:
10003:
9987:
9984:
9983:
9972:
9968:
9967:
9947:
9944:
9943:
9927:
9924:
9923:
9897:
9894:
9893:
9878:
9868:
9862:
9856:
9832:
9824:
9810:
9807:
9806:
9805:radians), then
9787:
9784:
9783:
9763:
9760:
9759:
9735:
9731:
9723:
9721:
9713:
9710:
9709:
9687:
9683:
9672:
9670:
9662:
9659:
9658:
9634:
9631:
9630:
9606:
9602:
9594:
9592:
9584:
9581:
9580:
9558:
9554:
9543:
9541:
9533:
9530:
9529:
9509:
9506:
9505:
9474:
9471:
9470:
9450:
9447:
9446:
9413:
9410:
9409:
9383:
9380:
9379:
9350:
9347:
9346:
9315:
9312:
9311:
9292:
9289:
9288:
9257:
9254:
9253:
9233:
9230:
9229:
9213:
9210:
9209:
9189:
9186:
9185:
9165:
9162:
9161:
9152:represents the
9137:
9134:
9133:
9119:
9113:
9107:
9102:
9069:
9055:
9047:
9045:
9041:
9040:
9031:
9017:
9009:
9007:
9003:
9002:
8996:
8992:
8983:
8969:
8961:
8959:
8955:
8954:
8952:
8944:
8940:
8939:
8932:
8928:
8927:
8921:
8918:
8917:
8886:
8872:
8864:
8862:
8858:
8857:
8845:
8841:
8835:
8831:
8822:
8808:
8800:
8798:
8794:
8793:
8787:
8783:
8774:
8760:
8752:
8750:
8746:
8745:
8743:
8735:
8731:
8730:
8723:
8719:
8718:
8712:
8709:
8708:
8686:
8674:
8670:
8669:
8657:
8653:
8647:
8643:
8634:
8622:
8618:
8617:
8611:
8607:
8598:
8586:
8582:
8581:
8572:
8560:
8556:
8555:
8544:
8539:
8538:
8529:
8524:
8523:
8522:
8518:
8509:
8497:
8493:
8492:
8481:
8476:
8475:
8466:
8461:
8460:
8459:
8455:
8441:
8437:
8433:
8429:
8418:
8413:
8412:
8403:
8398:
8397:
8396:
8392:
8390:
8387:
8386:
8370:
8367:
8366:
8350:
8347:
8346:
8329:
8324:
8323:
8314:
8309:
8308:
8306:
8303:
8302:
8277:
8271:
8266:
8265:
8254:
8248:
8243:
8242:
8231:
8225:
8220:
8219:
8208:
8200:
8192:
8189:
8188:
8077:
8075:
8072:
8071:
8055:
8052:
8051:
8035:
8032:
8031:
8015:
8012:
8011:
7935:
7933:
7930:
7929:
7904:
7901:
7900:
7869:
7866:
7865:
7834:
7830:
7821:
7807:
7799:
7797:
7793:
7792:
7790:
7779:
7775:
7768:
7758:
7754:
7747:
7725:
7711:
7703:
7701:
7697:
7696:
7690:
7686:
7677:
7663:
7655:
7653:
7649:
7648:
7646:
7638:
7634:
7633:
7626:
7622:
7621:
7615:
7612:
7611:
7589:
7577:
7573:
7572:
7566:
7562:
7553:
7541:
7537:
7536:
7527:
7515:
7511:
7510:
7499:
7494:
7493:
7484:
7479:
7478:
7477:
7473:
7462:
7454:
7443:
7438:
7437:
7428:
7423:
7422:
7421:
7417:
7405:
7393:
7389:
7388:
7376:
7372:
7371:
7361:
7357:
7356:
7355:
7351:
7349:
7346:
7345:
7321:
7315:
7310:
7309:
7298:
7292:
7287:
7286:
7275:
7267:
7259:
7256:
7255:
7213:
7205:
7204:
7200:
7199:
7198:
7194:
7192:
7189:
7188:
7116:
7114:
7111:
7110:
7047:
7045:
7042:
7041:
7038:
7010:
7006:
7004:
7001:
7000:
6977:
6973:
6971:
6968:
6967:
6939:
6935:
6931:
6930:
6916:
6912:
6908:
6907:
6898:
6894:
6892:
6890:
6887:
6886:
6862:
6858:
6856:
6853:
6852:
6833:
6821:
6817:
6816:
6810:
6806:
6795:
6787:
6781:
6777:
6765:
6753:
6749:
6748:
6742:
6738:
6724:
6718:
6713:
6712:
6701:
6695:
6690:
6689:
6670:
6664:
6659:
6658:
6647:
6641:
6636:
6635:
6634:
6630:
6628:
6625:
6624:
6598:
6592:
6587:
6586:
6575:
6569:
6564:
6563:
6553:
6544:
6536:
6530:
6529:
6528:
6519:
6514:
6513:
6504:
6499:
6498:
6484:
6476:
6468:
6465:
6464:
6416:
6408:
6407:
6403:
6402:
6401:
6397:
6395:
6392:
6391:
6330:
6328:
6325:
6324:
6293:
6291:
6288:
6287:
6284:
6269:
6267:
6262:
6259:
6256:
6253:
6251:
6241:
6239:
6216:
6203:
6196:
6195:
6186:
6179:
6175:
6169:
6168:
6156:
6153:
6152:
6145:
6142:
6139:
6136:
6134:
6104:
6100:
6085:
6083:
6073:
6066:
6065:
6056:
6049:
6045:
6039:
6036:
6035:
6008:
6004:
5997:
5993:
5987:
5986:
5974:
5970:
5959:
5942:
5939:
5938:
5919:
5915:
5907:
5901:
5900:
5880:
5872:
5869:
5868:
5844:
5837:
5826:
5819:
5815:
5811:
5803:
5800:
5799:
5777:
5773:
5765:
5757:
5754:
5753:
5745:
5665:
5651:
5643:
5641:
5637:
5636:
5628:
5622:
5617:
5605:
5602:
5601:
5568:
5554:
5546:
5544:
5540:
5539:
5531:
5520:
5516:
5504:
5500:
5495:
5493:
5490:
5489:
5449:
5446:
5445:
5423:
5420:
5419:
5399:
5396:
5395:
5361:
5358:
5357:
5340:
5335:
5334:
5332:
5329:
5328:
5313:
5307:
5289:
5288:
5264:
5263:
5258:
5242:
5241:
5224:
5218:
5217:
5211:
5206:
5193:
5192:
5187:
5179:
5152:
5146:
5145:
5119:
5113:
5112:
5106:
5101:
5088:
5087:
5072:
5071:
5054:
5028:
5022:
5021:
5015:
5010:
4988:
4987:
4970:
4964:
4963:
4957:
4952:
4941:
4925:
4923:
4920:
4919:
4898:
4895:
4894:
4877:
4872:
4871:
4841:
4837:
4823:
4820:
4819:
4763:
4760:
4759:
4740:
4737:
4736:
4705:
4702:
4701:
4678:
4674:
4659:
4655:
4640:
4636:
4627:
4623:
4615:
4612:
4611:
4588:
4587:
4572:
4568:
4550:
4546:
4534:
4533:
4527:
4516:
4492:
4489:
4488:
4451:
4448:
4447:
4430:
4425:
4424:
4398:
4395:
4394:
4359:
4356:
4355:
4322:
4321:
4317:
4315:
4312:
4311:
4276:
4272:
4261:
4260:
4256:
4250:
4239:
4212:
4196:
4192:
4174:
4170:
4163:
4161:
4157:
4151:
4140:
4134:
4131:
4130:
4114:
4111:
4110:
4094:
4091:
4090:
4068:
4065:
4064:
4033:
4030:
4029:
4005:
4002:
4001:
4000:. In the limit
3973:
3950:
3947:
3946:
3900:
3897:
3896:
3865:
3861:
3850:
3849:
3845:
3839:
3834:
3813:
3809:
3798:
3797:
3793:
3791:
3788:
3787:
3740:
3736:
3725:
3724:
3720:
3685:
3681:
3672:
3668:
3647:
3643:
3632:
3626:
3621:
3616:
3612:
3611:
3607:
3601:
3590:
3578:
3575:
3574:
3538:
3535:
3534:
3506:
3483:
3480:
3479:
3443:
3439:
3428:
3427:
3423:
3397:
3393:
3384:
3380:
3359:
3355:
3344:
3343:
3339:
3338:
3334:
3332:
3329:
3328:
3297:
3293:
3282:
3281:
3277:
3271:
3266:
3245:
3241:
3230:
3229:
3225:
3223:
3220:
3219:
3203:
3202:
3179:
3175:
3164:
3163:
3159:
3133:
3129:
3120:
3116:
3095:
3091:
3080:
3079:
3075:
3069:
3064:
3054:
3043:
3023:
3022:
2997:
2993:
2982:
2981:
2977:
2971:
2966:
2927:
2923:
2914:
2910:
2889:
2885:
2874:
2873:
2869:
2863:
2858:
2853:
2849:
2843:
2832:
2812:
2811:
2786:
2782:
2771:
2770:
2766:
2760:
2755:
2716:
2712:
2703:
2699:
2678:
2674:
2663:
2657:
2652:
2647:
2643:
2642:
2638:
2632:
2621:
2606:
2604:
2601:
2600:
2562:
2558:
2547:
2546:
2542:
2536:
2525:
2480:
2476:
2467:
2463:
2442:
2438:
2427:
2421:
2416:
2411:
2407:
2401:
2390:
2384:
2381:
2380:
2341:
2337:
2326:
2325:
2321:
2315:
2304:
2277:
2261:
2257:
2239:
2235:
2228:
2226:
2222:
2216:
2205:
2199:
2196:
2195:
2173:
2171:
2168:
2167:
2133:
2130:
2129:
2106:
2102:
2093:
2089:
2078:
2075:
2074:
2038:
2034:
2023:
2022:
2018:
1992:
1988:
1979:
1975:
1954:
1950:
1939:
1938:
1934:
1933:
1929:
1927:
1924:
1923:
1889:
1886:
1885:
1869:
1866:
1865:
1840:
1837:
1836:
1796:
1793:
1792:
1769:
1765:
1763:
1760:
1759:
1716:
1712:
1703:
1699:
1678:
1674:
1663:
1657:
1652:
1635:
1619:
1615:
1597:
1593:
1586:
1584:
1582:
1579:
1578:
1529:
1525:
1516:
1512:
1498:
1495:
1494:
1474:
1470:
1455:
1451:
1446:
1443:
1442:
1408:
1405:
1404:
1378:
1374:
1365:
1361:
1340:
1336:
1328:
1325:
1324:
1286:
1282:
1273:
1269:
1248:
1244:
1233:
1227:
1222:
1183:
1175:
1171:
1170:
1157:
1153:
1152:
1130:
1126:
1108:
1104:
1096:
1093:
1092:
1054:
1053:
1036:
1030:
1029:
1023:
1018:
991:
975:
971:
953:
949:
942:
940:
936:
930:
919:
903:
890:
889:
874:
870:
852:
848:
836:
835:
829:
818:
802:
781:
778:
777:
732:
729:
728:
705:
701:
692:
688:
669:
667:
656:
653:
652:
615:
582:
578:
576:
573:
572:
553:
552:
537:
533:
515:
511:
499:
498:
492:
481:
465:
444:
441:
440:
410:
407:
406:
386:
383:
382:
357:
352:
351:
325:
322:
321:
318:
312:
304:signed distance
268:
265:
264:
240:
237:
236:
161:
155:in some cases.
128:
123:
122:
96:
93:
92:
72:is also called
28:
23:
22:
15:
12:
11:
5:
13221:
13211:
13210:
13205:
13200:
13195:
13178:
13177:
13175:
13174:
13169:
13164:
13159:
13154:
13152:Gabriel's horn
13149:
13144:
13143:
13142:
13137:
13132:
13127:
13122:
13114:
13113:
13112:
13103:
13101:
13097:
13096:
13093:
13092:
13090:
13089:
13084:
13082:List of limits
13078:
13075:
13074:
13072:
13071:
13070:
13069:
13064:
13059:
13049:
13048:
13047:
13037:
13032:
13027:
13022:
13016:
13014:
13005:
13001:
13000:
12998:
12997:
12990:
12983:
12981:Leonhard Euler
12978:
12973:
12968:
12963:
12958:
12953:
12948:
12943:
12938:
12933:
12927:
12925:
12919:
12918:
12916:
12915:
12910:
12905:
12900:
12895:
12889:
12887:
12881:
12880:
12878:
12877:
12876:
12875:
12870:
12865:
12860:
12855:
12850:
12845:
12840:
12835:
12830:
12822:
12821:
12820:
12815:
12814:
12813:
12808:
12798:
12793:
12788:
12783:
12778:
12773:
12765:
12759:
12757:
12753:
12752:
12750:
12749:
12748:
12747:
12742:
12737:
12732:
12724:
12719:
12714:
12709:
12704:
12699:
12694:
12689:
12684:
12682:Hessian matrix
12679:
12674:
12668:
12666:
12660:
12659:
12657:
12656:
12655:
12654:
12649:
12644:
12639:
12637:Line integrals
12631:
12630:
12629:
12624:
12619:
12614:
12609:
12600:
12598:
12592:
12591:
12589:
12588:
12583:
12578:
12577:
12576:
12571:
12563:
12558:
12557:
12556:
12546:
12545:
12544:
12539:
12534:
12524:
12519:
12518:
12517:
12507:
12502:
12497:
12492:
12487:
12485:Antiderivative
12481:
12479:
12473:
12472:
12470:
12469:
12468:
12467:
12462:
12457:
12447:
12446:
12445:
12440:
12432:
12431:
12430:
12425:
12420:
12415:
12405:
12404:
12403:
12398:
12393:
12388:
12380:
12379:
12378:
12373:
12372:
12371:
12361:
12356:
12351:
12346:
12341:
12331:
12330:
12329:
12324:
12314:
12309:
12304:
12299:
12294:
12289:
12283:
12281:
12275:
12274:
12272:
12271:
12266:
12261:
12256:
12255:
12254:
12244:
12238:
12236:
12230:
12229:
12227:
12226:
12221:
12216:
12211:
12206:
12201:
12196:
12191:
12186:
12181:
12176:
12171:
12166:
12160:
12158:
12152:
12151:
12144:
12143:
12136:
12129:
12121:
12115:
12114:
12108:
12098:
12090:
12085:
12071:
12052:
12047:
12027:
12026:External links
12024:
12023:
12022:
12016:
12001:
11998:
11995:
11994:
11977:
11960:
11943:
11905:
11878:(3): 162–171.
11862:
11850:
11831:
11824:
11798:
11757:
11750:
11723:
11722:
11720:
11717:
11716:
11715:
11710:
11705:
11700:
11695:
11690:
11685:
11680:
11675:
11670:
11665:
11663:Arc (geometry)
11658:
11655:
11622:
11619:
11599:
11588:
11587:
11576:
11571:
11568:
11565:
11562:
11558:
11554:
11551:
11548:
11545:
11541:
11538:
11523:
11522:
11510:
11507:
11499:
11496:
11491:
11488:
11485:
11482:
11479:
11476:
11471:
11467:
11463:
11454:
11451:
11446:
11443:
11440:
11437:
11434:
11431:
11426:
11422:
11418:
11412:
11409:
11406:
11403:
11400:
11397:
11394:
11391:
11388:
11383:
11380:
11376:
11370:
11365:
11362:
11359:
11356:
11353:
11349:
11345:
11338:
11333:
11329:
11325:
11322:
11319:
11316:
11313:
11290:
11279:
11278:
11266:
11263:
11258:
11255:
11252:
11249:
11244:
11238:
11234:
11231:
11228:
11224:
11221:
11216:
11211:
11205:
11200:
11196:
11192:
11189:
11186:
11183:
11180:
11157:
11148:The length of
11146:
11145:
11144:
11143:
11131:
11127:
11124:
11121:
11118:
11115:
11110:
11106:
11102:
11099:
11096:
11093:
11090:
11074:
11073:
11072:
11071:
11060:
11057:
11054:
11051:
11048:
11045:
11042:
11038:
11035:
11032:
11029:
11026:
11021:
11017:
11013:
11010:
11007:
11004:
11001:
10998:
10993:
10989:
10985:
10982:
10979:
10976:
10973:
10970:
10945:
10925:
10905:
10902:
10899:
10896:
10893:
10890:
10887:
10884:
10881:
10868:the (pseudo-)
10857:
10833:
10821:
10818:
10766:
10763:
10759:
10755:
10752:
10749:
10746:
10743:
10740:
10731:The graph of
10710:
10707:
10703:
10702:
10691:
10685:
10680:
10677:
10672:
10669:
10664:
10661:
10658:
10655:
10641:
10640:
10624:
10620:
10615:
10612:
10607:
10604:
10600:
10594:
10590:
10586:
10583:
10581:
10579:
10574:
10570:
10566:
10561:
10558:
10553:
10548:
10544:
10540:
10537:
10535:
10533:
10528:
10524:
10520:
10517:
10512:
10508:
10504:
10501:
10498:
10496:
10492:
10488:
10484:
10481:
10480:
10464:, he used the
10434:
10433:
10422:
10419:
10416:
10413:
10410:
10407:
10404:
10401:
10398:
10395:
10392:
10386:
10383:
10378:
10372:
10369:
10364:
10361:
10347:
10346:
10332:
10329:
10324:
10318:
10315:
10283:
10282:
10267:
10264:
10259:
10255:
10252:
10206:
10203:
10147:
10144:
10110:
10107:
10105:
10102:
10101:
10100:
10095:
10090:
10085:
10080:
10075:
10070:
10063:
10060:
10028:
10027:
10011:
9991:
9980:
9951:
9931:
9907:
9904:
9901:
9858:Main article:
9855:
9852:
9851:
9850:
9849:
9848:
9831:
9827:
9823:
9820:
9817:
9814:
9794:
9791:
9767:
9756:
9745:
9734:
9729:
9726:
9720:
9717:
9697:
9686:
9681:
9678:
9675:
9669:
9666:
9638:
9627:
9616:
9609:
9605:
9600:
9597:
9591:
9588:
9568:
9561:
9557:
9552:
9549:
9546:
9540:
9537:
9513:
9502:
9490:
9487:
9484:
9481:
9478:
9454:
9440:
9429:
9426:
9423:
9420:
9417:
9402:
9390:
9387:
9366:
9363:
9360:
9357:
9354:
9334:
9331:
9328:
9325:
9322:
9319:
9296:
9276:
9273:
9270:
9267:
9264:
9261:
9237:
9217:
9193:
9169:
9141:
9109:Main article:
9106:
9103:
9101:
9098:
9086:
9083:
9080:
9072:
9067:
9061:
9058:
9053:
9050:
9044:
9039:
9034:
9029:
9023:
9020:
9015:
9012:
9006:
8999:
8995:
8991:
8986:
8981:
8975:
8972:
8967:
8964:
8958:
8947:
8943:
8935:
8931:
8926:
8903:
8900:
8897:
8889:
8884:
8878:
8875:
8870:
8867:
8861:
8856:
8853:
8848:
8844:
8838:
8834:
8830:
8825:
8820:
8814:
8811:
8806:
8803:
8797:
8790:
8786:
8782:
8777:
8772:
8766:
8763:
8758:
8755:
8749:
8738:
8734:
8726:
8722:
8717:
8694:
8689:
8684:
8680:
8677:
8673:
8668:
8665:
8660:
8656:
8650:
8646:
8642:
8637:
8632:
8628:
8625:
8621:
8614:
8610:
8606:
8601:
8596:
8592:
8589:
8585:
8580:
8575:
8570:
8566:
8563:
8559:
8553:
8547:
8542:
8537:
8532:
8527:
8521:
8517:
8512:
8507:
8503:
8500:
8496:
8490:
8484:
8479:
8474:
8469:
8464:
8458:
8454:
8450:
8444:
8440:
8436:
8432:
8427:
8421:
8416:
8411:
8406:
8401:
8395:
8374:
8354:
8332:
8327:
8322:
8317:
8312:
8287:
8283:
8280:
8274:
8269:
8264:
8260:
8257:
8251:
8246:
8241:
8237:
8234:
8228:
8223:
8218:
8215:
8211:
8207:
8203:
8199:
8196:
8174:
8171:
8168:
8165:
8162:
8159:
8156:
8153:
8150:
8147:
8144:
8141:
8138:
8135:
8132:
8129:
8126:
8123:
8120:
8117:
8114:
8111:
8108:
8105:
8102:
8099:
8096:
8093:
8090:
8087:
8084:
8080:
8059:
8039:
8019:
7999:
7996:
7993:
7990:
7987:
7984:
7981:
7978:
7975:
7972:
7969:
7966:
7963:
7960:
7957:
7954:
7951:
7948:
7945:
7942:
7938:
7914:
7911:
7908:
7888:
7885:
7882:
7879:
7876:
7873:
7851:
7848:
7845:
7837:
7833:
7829:
7824:
7819:
7813:
7810:
7805:
7802:
7796:
7787:
7782:
7778:
7774:
7771:
7766:
7761:
7757:
7753:
7750:
7746:
7742:
7739:
7736:
7728:
7723:
7717:
7714:
7709:
7706:
7700:
7693:
7689:
7685:
7680:
7675:
7669:
7666:
7661:
7658:
7652:
7641:
7637:
7629:
7625:
7620:
7597:
7592:
7587:
7583:
7580:
7576:
7569:
7565:
7561:
7556:
7551:
7547:
7544:
7540:
7535:
7530:
7525:
7521:
7518:
7514:
7508:
7502:
7497:
7492:
7487:
7482:
7476:
7472:
7468:
7465:
7460:
7457:
7452:
7446:
7441:
7436:
7431:
7426:
7420:
7416:
7413:
7408:
7403:
7399:
7396:
7392:
7386:
7379:
7375:
7370:
7364:
7360:
7354:
7331:
7327:
7324:
7318:
7313:
7308:
7304:
7301:
7295:
7290:
7285:
7282:
7278:
7274:
7270:
7266:
7263:
7243:
7239:
7235:
7232:
7229:
7225:
7221:
7216:
7212:
7208:
7203:
7197:
7174:
7171:
7168:
7165:
7162:
7159:
7156:
7153:
7150:
7147:
7144:
7141:
7138:
7135:
7132:
7129:
7126:
7123:
7119:
7096:
7093:
7090:
7087:
7084:
7081:
7078:
7075:
7072:
7069:
7066:
7063:
7060:
7057:
7054:
7050:
7037:
7034:
7021:
7018:
7013:
7009:
6988:
6985:
6980:
6976:
6951:
6947:
6942:
6938:
6934:
6928:
6924:
6919:
6915:
6911:
6904:
6901:
6897:
6868:
6865:
6861:
6836:
6831:
6827:
6824:
6820:
6813:
6809:
6805:
6801:
6798:
6793:
6790:
6784:
6780:
6776:
6773:
6768:
6763:
6759:
6756:
6752:
6745:
6741:
6737:
6734:
6730:
6727:
6721:
6716:
6711:
6707:
6704:
6698:
6693:
6688:
6685:
6681:
6676:
6673:
6667:
6662:
6657:
6653:
6650:
6644:
6639:
6633:
6608:
6604:
6601:
6595:
6590:
6585:
6581:
6578:
6572:
6567:
6562:
6556:
6550:
6547:
6542:
6539:
6533:
6527:
6522:
6517:
6507:
6502:
6497:
6494:
6491:
6487:
6483:
6479:
6475:
6472:
6446:
6442:
6438:
6435:
6432:
6428:
6424:
6419:
6415:
6411:
6406:
6400:
6379:
6376:
6373:
6370:
6367:
6364:
6361:
6358:
6355:
6352:
6349:
6346:
6343:
6340:
6337:
6333:
6312:
6309:
6306:
6303:
6300:
6296:
6283:
6280:
6223:
6220:
6215:
6210:
6206:
6200:
6193:
6189:
6183:
6178:
6172:
6166:
6163:
6160:
6116:
6107:
6103:
6099:
6096:
6091:
6088:
6080:
6076:
6070:
6063:
6059:
6053:
6048:
6044:
6021:
6017:
6011:
6007:
6003:
6000:
5996:
5990:
5985:
5982:
5977:
5973:
5969:
5966:
5962:
5958:
5955:
5952:
5949:
5946:
5922:
5918:
5914:
5911:
5904:
5899:
5896:
5893:
5890:
5887:
5883:
5879:
5876:
5855:
5851:
5847:
5841:
5836:
5833:
5829:
5823:
5818:
5814:
5810:
5807:
5787:
5780:
5776:
5772:
5769:
5764:
5761:
5744:
5741:
5682:
5679:
5676:
5668:
5663:
5657:
5654:
5649:
5646:
5640:
5635:
5632:
5625:
5620:
5616:
5612:
5609:
5585:
5582:
5579:
5571:
5566:
5560:
5557:
5552:
5549:
5543:
5538:
5535:
5530:
5523:
5519:
5515:
5512:
5507:
5503:
5499:
5471:
5468:
5465:
5462:
5459:
5456:
5453:
5433:
5430:
5427:
5403:
5383:
5380:
5377:
5374:
5371:
5368:
5365:
5343:
5338:
5319:Quarter circle
5306:
5303:
5287:
5284:
5281:
5278:
5275:
5272:
5269:
5267:
5265:
5256:
5253:
5245:
5240:
5237:
5234:
5230:
5227:
5221:
5214:
5209:
5205:
5201:
5198:
5196:
5194:
5186:
5177:
5174:
5168:
5165:
5162:
5158:
5155:
5149:
5144:
5141:
5138:
5135:
5132:
5129:
5125:
5122:
5116:
5109:
5104:
5100:
5096:
5093:
5091:
5089:
5086:
5083:
5075:
5070:
5067:
5064:
5060:
5057:
5053:
5050:
5047:
5044:
5041:
5038:
5034:
5031:
5025:
5018:
5013:
5009:
5005:
5002:
4999:
4991:
4986:
4983:
4980:
4976:
4973:
4967:
4960:
4955:
4951:
4947:
4944:
4942:
4940:
4937:
4934:
4931:
4928:
4927:
4905:
4902:
4880:
4875:
4870:
4867:
4864:
4861:
4858:
4855:
4852:
4847:
4844:
4840:
4836:
4833:
4830:
4827:
4803:
4800:
4797:
4794:
4791:
4788:
4785:
4782:
4779:
4776:
4773:
4770:
4767:
4744:
4724:
4721:
4718:
4715:
4712:
4709:
4689:
4686:
4681:
4677:
4673:
4668:
4665:
4662:
4658:
4654:
4651:
4648:
4643:
4639:
4635:
4630:
4626:
4622:
4619:
4591:
4586:
4581:
4578:
4575:
4571:
4567:
4564:
4561:
4558:
4553:
4549:
4545:
4542:
4537:
4530:
4525:
4522:
4519:
4515:
4511:
4508:
4505:
4502:
4499:
4496:
4467:
4464:
4461:
4458:
4455:
4433:
4428:
4423:
4420:
4417:
4414:
4411:
4408:
4405:
4402:
4378:
4375:
4372:
4369:
4366:
4363:
4342:
4338:
4335:
4332:
4328:
4325:
4320:
4295:
4292:
4288:
4284:
4279:
4275:
4271:
4267:
4264:
4259:
4253:
4248:
4245:
4242:
4238:
4234:
4231:
4228:
4224:
4218:
4215:
4210:
4205:
4202:
4199:
4195:
4191:
4188:
4185:
4182:
4177:
4173:
4169:
4166:
4160:
4154:
4149:
4146:
4143:
4139:
4118:
4098:
4078:
4075:
4072:
4052:
4049:
4046:
4043:
4040:
4037:
4018:
4015:
4012:
4009:
3989:
3986:
3983:
3980:
3976:
3972:
3969:
3966:
3963:
3960:
3957:
3954:
3934:
3931:
3928:
3925:
3922:
3919:
3916:
3913:
3910:
3907:
3904:
3884:
3881:
3877:
3873:
3868:
3864:
3860:
3856:
3853:
3848:
3842:
3837:
3833:
3829:
3825:
3821:
3816:
3812:
3808:
3804:
3801:
3796:
3773:
3770:
3767:
3764:
3761:
3757:
3752:
3748:
3743:
3739:
3735:
3731:
3728:
3723:
3719:
3715:
3711:
3708:
3702:
3699:
3694:
3691:
3688:
3684:
3680:
3675:
3671:
3667:
3664:
3661:
3656:
3653:
3650:
3646:
3642:
3638:
3635:
3629:
3624:
3620:
3615:
3610:
3604:
3599:
3596:
3593:
3589:
3585:
3582:
3560:
3557:
3554:
3551:
3548:
3545:
3542:
3522:
3519:
3516:
3513:
3509:
3505:
3502:
3499:
3496:
3493:
3490:
3487:
3467:
3464:
3460:
3455:
3451:
3446:
3442:
3438:
3434:
3431:
3426:
3422:
3418:
3414:
3411:
3406:
3403:
3400:
3396:
3392:
3387:
3383:
3379:
3376:
3373:
3368:
3365:
3362:
3358:
3354:
3350:
3347:
3342:
3337:
3316:
3313:
3309:
3305:
3300:
3296:
3292:
3288:
3285:
3280:
3274:
3269:
3265:
3261:
3257:
3253:
3248:
3244:
3240:
3236:
3233:
3228:
3201:
3198:
3191:
3187:
3182:
3178:
3174:
3170:
3167:
3162:
3158:
3154:
3150:
3147:
3142:
3139:
3136:
3132:
3128:
3123:
3119:
3115:
3112:
3109:
3104:
3101:
3098:
3094:
3090:
3086:
3083:
3078:
3072:
3067:
3063:
3057:
3052:
3049:
3046:
3042:
3038:
3035:
3032:
3028:
3026:
3024:
3020:
3016:
3013:
3009:
3005:
3000:
2996:
2992:
2988:
2985:
2980:
2974:
2969:
2965:
2961:
2958:
2955:
2948:
2944:
2941:
2936:
2933:
2930:
2926:
2922:
2917:
2913:
2909:
2906:
2903:
2898:
2895:
2892:
2888:
2884:
2880:
2877:
2872:
2866:
2861:
2857:
2852:
2846:
2841:
2838:
2835:
2831:
2827:
2824:
2821:
2817:
2815:
2813:
2809:
2805:
2802:
2798:
2794:
2789:
2785:
2781:
2777:
2774:
2769:
2763:
2758:
2754:
2750:
2746:
2742:
2739:
2733:
2730:
2725:
2722:
2719:
2715:
2711:
2706:
2702:
2698:
2695:
2692:
2687:
2684:
2681:
2677:
2673:
2669:
2666:
2660:
2655:
2651:
2646:
2641:
2635:
2630:
2627:
2624:
2620:
2616:
2613:
2610:
2608:
2584:
2581:
2578:
2574:
2570:
2565:
2561:
2557:
2553:
2550:
2545:
2539:
2534:
2531:
2528:
2524:
2520:
2517:
2514:
2510:
2506:
2503:
2497:
2494:
2489:
2486:
2483:
2479:
2475:
2470:
2466:
2462:
2459:
2456:
2451:
2448:
2445:
2441:
2437:
2433:
2430:
2424:
2419:
2415:
2410:
2404:
2399:
2396:
2393:
2389:
2375:
2374:
2363:
2360:
2357:
2353:
2349:
2344:
2340:
2336:
2332:
2329:
2324:
2318:
2313:
2310:
2307:
2303:
2299:
2296:
2293:
2289:
2283:
2280:
2275:
2270:
2267:
2264:
2260:
2256:
2253:
2250:
2247:
2242:
2238:
2234:
2231:
2225:
2219:
2214:
2211:
2208:
2204:
2182:
2179:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2115:
2112:
2109:
2105:
2101:
2096:
2092:
2088:
2085:
2082:
2062:
2059:
2055:
2050:
2046:
2041:
2037:
2033:
2029:
2026:
2021:
2017:
2013:
2009:
2006:
2001:
1998:
1995:
1991:
1987:
1982:
1978:
1974:
1971:
1968:
1963:
1960:
1957:
1953:
1949:
1945:
1942:
1937:
1932:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1873:
1853:
1850:
1847:
1844:
1812:
1809:
1806:
1803:
1800:
1779:
1775:
1772:
1768:
1756:
1745:
1742:
1739:
1733:
1730:
1725:
1722:
1719:
1715:
1711:
1706:
1702:
1698:
1695:
1692:
1687:
1684:
1681:
1677:
1673:
1669:
1666:
1660:
1655:
1651:
1647:
1641:
1638:
1633:
1628:
1625:
1622:
1618:
1614:
1611:
1608:
1605:
1600:
1596:
1592:
1589:
1566:
1563:
1559:
1556:
1553:
1550:
1547:
1543:
1538:
1535:
1532:
1528:
1524:
1519:
1515:
1511:
1508:
1505:
1502:
1482:
1477:
1473:
1469:
1464:
1461:
1458:
1454:
1450:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1392:
1387:
1384:
1381:
1377:
1373:
1368:
1364:
1360:
1357:
1354:
1349:
1346:
1343:
1339:
1335:
1332:
1312:
1309:
1303:
1300:
1295:
1292:
1289:
1285:
1281:
1276:
1272:
1268:
1265:
1262:
1257:
1254:
1251:
1247:
1243:
1239:
1236:
1230:
1225:
1221:
1217:
1214:
1211:
1208:
1205:
1199:
1196:
1193:
1189:
1186:
1178:
1174:
1166:
1163:
1160:
1156:
1151:
1147:
1144:
1139:
1136:
1133:
1129:
1125:
1122:
1119:
1116:
1111:
1107:
1103:
1100:
1071:
1068:
1065:
1057:
1052:
1049:
1046:
1042:
1039:
1033:
1026:
1021:
1017:
1013:
1010:
1007:
1003:
997:
994:
989:
984:
981:
978:
974:
970:
967:
964:
961:
956:
952:
948:
945:
939:
933:
928:
925:
922:
918:
912:
909:
906:
902:
898:
893:
888:
883:
880:
877:
873:
869:
866:
863:
860:
855:
851:
847:
844:
839:
832:
827:
824:
821:
817:
811:
808:
805:
801:
797:
794:
791:
788:
785:
763:
760:
757:
754:
751:
748:
745:
742:
739:
736:
714:
711:
708:
704:
700:
695:
691:
687:
682:
678:
675:
672:
666:
663:
660:
640:
637:
634:
631:
628:
625:
622:
618:
614:
611:
608:
605:
602:
599:
596:
593:
590:
585:
581:
556:
551:
546:
543:
540:
536:
532:
529:
526:
523:
518:
514:
510:
507:
502:
495:
490:
487:
484:
480:
474:
471:
468:
464:
460:
457:
454:
451:
448:
426:
423:
420:
417:
414:
390:
360:
355:
350:
347:
344:
341:
338:
335:
332:
329:
311:
308:
272:
262:
258:
244:
197:polygonal path
160:
157:
147:The advent of
131:
126:
121:
118:
115:
112:
109:
106:
103:
100:
26:
9:
6:
4:
3:
2:
13220:
13209:
13206:
13204:
13201:
13199:
13196:
13194:
13191:
13190:
13188:
13173:
13170:
13168:
13165:
13163:
13160:
13158:
13155:
13153:
13150:
13148:
13145:
13141:
13138:
13136:
13133:
13131:
13128:
13126:
13123:
13121:
13118:
13117:
13115:
13111:
13108:
13107:
13105:
13104:
13102:
13098:
13088:
13085:
13083:
13080:
13079:
13076:
13068:
13065:
13063:
13060:
13058:
13055:
13054:
13053:
13050:
13046:
13043:
13042:
13041:
13038:
13036:
13033:
13031:
13028:
13026:
13023:
13021:
13018:
13017:
13015:
13013:
13009:
13006:
13002:
12996:
12995:
12991:
12989:
12988:
12984:
12982:
12979:
12977:
12974:
12972:
12969:
12967:
12964:
12962:
12959:
12957:
12956:Infinitesimal
12954:
12952:
12949:
12947:
12944:
12942:
12939:
12937:
12934:
12932:
12929:
12928:
12926:
12924:
12920:
12914:
12911:
12909:
12906:
12904:
12901:
12899:
12896:
12894:
12891:
12890:
12888:
12882:
12874:
12871:
12869:
12866:
12864:
12861:
12859:
12856:
12854:
12851:
12849:
12846:
12844:
12841:
12839:
12836:
12834:
12831:
12829:
12826:
12825:
12823:
12819:
12816:
12812:
12809:
12807:
12804:
12803:
12802:
12799:
12797:
12794:
12792:
12789:
12787:
12784:
12782:
12779:
12777:
12774:
12772:
12769:
12768:
12766:
12764:
12761:
12760:
12758:
12754:
12746:
12743:
12741:
12738:
12736:
12733:
12731:
12728:
12727:
12725:
12723:
12720:
12718:
12715:
12713:
12710:
12708:
12705:
12703:
12700:
12698:
12697:Line integral
12695:
12693:
12690:
12688:
12685:
12683:
12680:
12678:
12675:
12673:
12670:
12669:
12667:
12665:
12661:
12653:
12650:
12648:
12645:
12643:
12640:
12638:
12635:
12634:
12632:
12628:
12625:
12623:
12620:
12618:
12615:
12613:
12610:
12608:
12605:
12604:
12602:
12601:
12599:
12597:
12593:
12587:
12584:
12582:
12579:
12575:
12572:
12570:
12569:Washer method
12567:
12566:
12564:
12562:
12559:
12555:
12552:
12551:
12550:
12547:
12543:
12540:
12538:
12535:
12533:
12532:trigonometric
12530:
12529:
12528:
12525:
12523:
12520:
12516:
12513:
12512:
12511:
12508:
12506:
12503:
12501:
12498:
12496:
12493:
12491:
12488:
12486:
12483:
12482:
12480:
12478:
12474:
12466:
12463:
12461:
12458:
12456:
12453:
12452:
12451:
12448:
12444:
12441:
12439:
12436:
12435:
12433:
12429:
12426:
12424:
12421:
12419:
12416:
12414:
12411:
12410:
12409:
12406:
12402:
12401:Related rates
12399:
12397:
12394:
12392:
12389:
12387:
12384:
12383:
12381:
12377:
12374:
12370:
12367:
12366:
12365:
12362:
12360:
12357:
12355:
12352:
12350:
12347:
12345:
12342:
12340:
12337:
12336:
12335:
12332:
12328:
12325:
12323:
12320:
12319:
12318:
12315:
12313:
12310:
12308:
12305:
12303:
12300:
12298:
12295:
12293:
12290:
12288:
12285:
12284:
12282:
12280:
12276:
12270:
12267:
12265:
12262:
12260:
12257:
12253:
12250:
12249:
12248:
12245:
12243:
12240:
12239:
12237:
12235:
12231:
12225:
12222:
12220:
12217:
12215:
12212:
12210:
12207:
12205:
12202:
12200:
12197:
12195:
12192:
12190:
12187:
12185:
12182:
12180:
12177:
12175:
12172:
12170:
12167:
12165:
12162:
12161:
12159:
12157:
12153:
12149:
12142:
12137:
12135:
12130:
12128:
12123:
12122:
12119:
12112:
12109:
12106:
12102:
12099:
12097:
12094:
12091:
12089:
12086:
12083:
12079:
12075:
12072:
12067:
12066:
12061:
12058:
12053:
12051:
12048:
12044:
12040:
12039:
12034:
12030:
12029:
12019:
12013:
12009:
12004:
12003:
11990:
11989:
11981:
11973:
11972:
11964:
11956:
11955:
11947:
11939:
11935:
11931:
11927:
11923:
11919:
11915:
11909:
11901:
11897:
11893:
11889:
11885:
11881:
11877:
11873:
11866:
11859:
11854:
11846:
11842:
11835:
11827:
11821:
11817:
11812:
11811:
11802:
11794:
11790:
11785:
11780:
11776:
11772:
11768:
11761:
11753:
11751:9780080955452
11747:
11743:
11738:
11737:
11728:
11724:
11714:
11711:
11709:
11706:
11704:
11701:
11699:
11698:Line integral
11696:
11694:
11691:
11689:
11686:
11684:
11681:
11679:
11676:
11674:
11671:
11669:
11668:Circumference
11666:
11664:
11661:
11660:
11654:
11652:
11648:
11644:
11640:
11635:
11620:
11617:
11597:
11574:
11566:
11560:
11556:
11552:
11546:
11539:
11536:
11528:
11527:
11526:
11508:
11505:
11497:
11494:
11483:
11477:
11469:
11465:
11461:
11452:
11449:
11438:
11432:
11424:
11420:
11416:
11401:
11395:
11389:
11381:
11378:
11374:
11368:
11363:
11360:
11357:
11354:
11351:
11347:
11343:
11336:
11331:
11327:
11323:
11317:
11311:
11304:
11303:
11302:
11288:
11264:
11261:
11253:
11247:
11229:
11222:
11219:
11203:
11198:
11194:
11190:
11184:
11178:
11171:
11170:
11169:
11155:
11125:
11119:
11113:
11108:
11100:
11094:
11088:
11081:
11080:
11079:
11078:
11077:
11055:
11052:
11049:
11043:
11040:
11036:
11027:
11019:
11015:
11011:
11008:
11005:
10999:
10991:
10987:
10980:
10974:
10968:
10961:
10960:
10959:
10958:
10957:
10943:
10923:
10903:
10894:
10891:
10888:
10882:
10879:
10871:
10870:metric tensor
10855:
10847:
10831:
10817:
10815:
10811:
10807:
10803:
10799:
10795:
10791:
10787:
10783:
10761:
10757:
10753:
10747:
10744:
10741:
10738:
10729:
10721:
10716:
10706:
10689:
10683:
10678:
10675:
10670:
10667:
10662:
10659:
10656:
10653:
10646:
10645:
10644:
10622:
10618:
10613:
10610:
10605:
10602:
10598:
10592:
10588:
10584:
10582:
10572:
10568:
10564:
10559:
10556:
10551:
10546:
10542:
10538:
10536:
10526:
10522:
10518:
10515:
10510:
10506:
10502:
10499:
10497:
10490:
10486:
10482:
10471:
10470:
10469:
10467:
10463:
10459:
10455:
10451:
10447:
10443:
10439:
10420:
10414:
10408:
10405:
10399:
10396:
10393:
10384:
10381:
10376:
10370:
10367:
10362:
10359:
10352:
10351:
10350:
10330:
10327:
10322:
10316:
10313:
10304:
10303:
10302:
10300:
10296:
10292:
10288:
10265:
10262:
10257:
10253:
10250:
10243:
10242:
10241:
10234:
10230:
10228:
10224:
10218:
10216:
10212:
10205:Integral form
10202:
10200:
10196:
10192:
10188:
10187:William Neile
10183:
10181:
10177:
10173:
10169:
10165:
10161:
10157:
10153:
10143:
10141:
10136:
10132:
10131:approximation
10128:
10124:
10120:
10116:
10099:
10096:
10094:
10091:
10089:
10086:
10084:
10081:
10079:
10076:
10074:
10071:
10069:
10066:
10065:
10059:
10045:
10025:
10009:
9989:
9981:
9978:
9965:
9949:
9929:
9921:
9920:
9919:
9905:
9902:
9899:
9891:
9890:great circles
9887:
9883:
9882:nautical mile
9877:
9873:
9867:
9861:
9829:
9825:
9821:
9818:
9815:
9812:
9792:
9789:
9781:
9765:
9757:
9743:
9732:
9727:
9724:
9718:
9715:
9695:
9684:
9679:
9676:
9673:
9667:
9664:
9656:
9652:
9636:
9628:
9614:
9607:
9603:
9598:
9595:
9589:
9586:
9566:
9559:
9555:
9550:
9547:
9544:
9538:
9535:
9527:
9511:
9503:
9488:
9485:
9482:
9479:
9476:
9468:
9452:
9444:
9443:
9441:
9427:
9424:
9421:
9418:
9415:
9407:
9403:
9401:
9388:
9385:
9364:
9361:
9358:
9355:
9352:
9332:
9329:
9326:
9323:
9320:
9317:
9310:
9309:
9308:
9294:
9274:
9271:
9268:
9265:
9262:
9259:
9251:
9235:
9215:
9207:
9206:circumference
9191:
9183:
9167:
9159:
9155:
9139:
9130:
9128:
9124:
9118:
9112:
9097:
9084:
9081:
9078:
9070:
9065:
9059:
9056:
9051:
9048:
9042:
9037:
9032:
9027:
9021:
9018:
9013:
9010:
9004:
8997:
8993:
8989:
8984:
8979:
8973:
8970:
8965:
8962:
8956:
8945:
8941:
8933:
8929:
8924:
8914:
8901:
8898:
8895:
8887:
8882:
8876:
8873:
8868:
8865:
8859:
8854:
8851:
8846:
8842:
8836:
8832:
8828:
8823:
8818:
8812:
8809:
8804:
8801:
8795:
8788:
8784:
8780:
8775:
8770:
8764:
8761:
8756:
8753:
8747:
8736:
8732:
8724:
8720:
8715:
8705:
8692:
8687:
8682:
8678:
8675:
8671:
8666:
8663:
8658:
8654:
8648:
8644:
8640:
8635:
8630:
8626:
8623:
8619:
8612:
8608:
8604:
8599:
8594:
8590:
8587:
8583:
8578:
8573:
8568:
8564:
8561:
8557:
8551:
8545:
8535:
8530:
8519:
8515:
8510:
8505:
8501:
8498:
8494:
8488:
8482:
8472:
8467:
8456:
8452:
8448:
8442:
8438:
8434:
8430:
8425:
8419:
8409:
8404:
8393:
8372:
8352:
8330:
8320:
8315:
8301:
8285:
8281:
8278:
8272:
8262:
8258:
8255:
8249:
8239:
8235:
8232:
8226:
8216:
8205:
8194:
8185:
8172:
8166:
8163:
8160:
8157:
8154:
8151:
8148:
8145:
8142:
8139:
8136:
8133:
8130:
8127:
8124:
8121:
8118:
8115:
8112:
8109:
8103:
8097:
8094:
8091:
8088:
8085:
8057:
8037:
8017:
7991:
7985:
7982:
7976:
7970:
7967:
7961:
7955:
7949:
7943:
7926:
7912:
7909:
7906:
7883:
7877:
7874:
7871:
7862:
7849:
7846:
7843:
7835:
7831:
7827:
7822:
7817:
7811:
7808:
7803:
7800:
7794:
7780:
7776:
7769:
7759:
7755:
7748:
7744:
7740:
7737:
7734:
7726:
7721:
7715:
7712:
7707:
7704:
7698:
7691:
7687:
7683:
7678:
7673:
7667:
7664:
7659:
7656:
7650:
7639:
7635:
7627:
7623:
7618:
7608:
7595:
7590:
7585:
7581:
7578:
7574:
7567:
7563:
7559:
7554:
7549:
7545:
7542:
7538:
7533:
7528:
7523:
7519:
7516:
7512:
7506:
7500:
7490:
7485:
7474:
7470:
7466:
7463:
7458:
7455:
7450:
7444:
7434:
7429:
7418:
7414:
7411:
7406:
7401:
7397:
7394:
7390:
7384:
7368:
7352:
7343:
7329:
7325:
7322:
7316:
7306:
7302:
7299:
7293:
7283:
7272:
7261:
7241:
7237:
7230:
7223:
7219:
7210:
7201:
7195:
7185:
7172:
7166:
7163:
7160:
7157:
7154:
7151:
7148:
7145:
7142:
7136:
7130:
7127:
7124:
7108:
7088:
7082:
7079:
7073:
7067:
7061:
7055:
7033:
7019:
7016:
7011:
7007:
6986:
6983:
6978:
6974:
6949:
6945:
6940:
6936:
6932:
6926:
6922:
6917:
6913:
6909:
6902:
6899:
6895:
6884:
6866:
6863:
6859:
6849:
6834:
6829:
6825:
6822:
6818:
6811:
6807:
6803:
6799:
6796:
6791:
6788:
6782:
6778:
6774:
6771:
6766:
6761:
6757:
6754:
6750:
6743:
6739:
6735:
6728:
6725:
6719:
6709:
6705:
6702:
6696:
6683:
6679:
6674:
6671:
6665:
6655:
6651:
6648:
6642:
6631:
6622:
6619:
6606:
6602:
6599:
6593:
6583:
6579:
6576:
6570:
6560:
6548:
6545:
6540:
6537:
6520:
6505:
6492:
6481:
6470:
6462:
6460:
6444:
6440:
6433:
6426:
6422:
6413:
6404:
6398:
6371:
6365:
6362:
6356:
6350:
6344:
6338:
6307:
6304:
6301:
6279:
6277:
6249:
6236:
6221:
6218:
6213:
6208:
6204:
6198:
6191:
6187:
6181:
6176:
6164:
6161:
6158:
6150:
6132:
6131:Gauss–Kronrod
6129:The 15-point
6127:
6114:
6105:
6101:
6097:
6094:
6089:
6086:
6078:
6074:
6068:
6061:
6057:
6051:
6046:
6042:
6033:
6019:
6015:
6009:
6005:
6001:
5998:
5994:
5983:
5980:
5975:
5967:
5964:
5960:
5956:
5953:
5947:
5944:
5920:
5916:
5912:
5909:
5897:
5894:
5891:
5888:
5885:
5881:
5877:
5874:
5853:
5849:
5845:
5839:
5834:
5831:
5827:
5821:
5816:
5812:
5808:
5805:
5798:The interval
5785:
5778:
5774:
5770:
5767:
5762:
5759:
5750:
5740:
5738:
5734:
5730:
5726:
5725:straight line
5722:
5718:
5714:
5710:
5706:
5702:
5698:
5693:
5680:
5677:
5674:
5666:
5661:
5655:
5652:
5647:
5644:
5638:
5633:
5630:
5623:
5618:
5614:
5610:
5607:
5599:
5596:
5583:
5580:
5577:
5569:
5564:
5558:
5555:
5550:
5547:
5541:
5536:
5533:
5528:
5521:
5517:
5513:
5510:
5505:
5501:
5497:
5487:
5485:
5469:
5463:
5457:
5454:
5451:
5431:
5428:
5425:
5417:
5401:
5381:
5375:
5369:
5366:
5363:
5341:
5326:
5317:
5312:
5302:
5285:
5279:
5273:
5270:
5268:
5254:
5251:
5235:
5228:
5225:
5212:
5207:
5203:
5199:
5197:
5184:
5175:
5172:
5163:
5156:
5153:
5136:
5130:
5123:
5120:
5107:
5102:
5098:
5094:
5092:
5084:
5081:
5065:
5058:
5055:
5045:
5039:
5032:
5029:
5016:
5011:
5007:
5003:
5000:
4997:
4981:
4974:
4971:
4958:
4953:
4949:
4945:
4943:
4935:
4929:
4917:
4903:
4900:
4878:
4862:
4859:
4856:
4850:
4845:
4842:
4838:
4834:
4831:
4828:
4825:
4817:
4798:
4795:
4792:
4780:
4777:
4774:
4768:
4765:
4756:
4742:
4722:
4716:
4713:
4710:
4687:
4684:
4679:
4675:
4671:
4666:
4663:
4660:
4656:
4652:
4649:
4646:
4641:
4637:
4633:
4628:
4624:
4620:
4617:
4609:
4604:
4579:
4576:
4573:
4569:
4562:
4559:
4551:
4547:
4540:
4528:
4523:
4520:
4517:
4513:
4506:
4500:
4494:
4486:
4483:
4481:
4462:
4459:
4456:
4431:
4415:
4412:
4409:
4403:
4400:
4392:
4376:
4370:
4367:
4364:
4340:
4333:
4326:
4323:
4318:
4309:
4293:
4286:
4277:
4273:
4265:
4262:
4257:
4251:
4246:
4243:
4240:
4236:
4232:
4229:
4222:
4216:
4203:
4200:
4197:
4193:
4186:
4183:
4175:
4171:
4164:
4158:
4152:
4147:
4144:
4141:
4137:
4116:
4096:
4076:
4070:
4050:
4041:
4035:
4016:
4007:
3984:
3978:
3974:
3967:
3964:
3961:
3955:
3952:
3929:
3926:
3923:
3917:
3914:
3911:
3905:
3902:
3882:
3879:
3875:
3866:
3862:
3854:
3851:
3846:
3840:
3835:
3831:
3827:
3823:
3814:
3810:
3802:
3799:
3794:
3784:
3771:
3765:
3762:
3759:
3755:
3750:
3741:
3737:
3729:
3726:
3721:
3717:
3713:
3709:
3706:
3692:
3689:
3686:
3682:
3678:
3673:
3669:
3662:
3659:
3654:
3651:
3648:
3644:
3636:
3633:
3627:
3622:
3618:
3613:
3608:
3602:
3597:
3594:
3591:
3587:
3583:
3572:
3571:, it becomes
3555:
3549:
3546:
3543:
3517:
3511:
3507:
3500:
3497:
3494:
3488:
3485:
3465:
3462:
3458:
3453:
3444:
3440:
3432:
3429:
3424:
3420:
3416:
3404:
3401:
3398:
3394:
3390:
3385:
3381:
3374:
3371:
3366:
3363:
3360:
3356:
3348:
3345:
3340:
3335:
3314:
3311:
3307:
3298:
3294:
3286:
3283:
3278:
3272:
3267:
3263:
3259:
3255:
3246:
3242:
3234:
3231:
3226:
3216:
3199:
3196:
3189:
3180:
3176:
3168:
3165:
3160:
3156:
3152:
3140:
3137:
3134:
3130:
3126:
3121:
3117:
3110:
3107:
3102:
3099:
3096:
3092:
3084:
3081:
3076:
3070:
3065:
3061:
3055:
3050:
3047:
3044:
3040:
3036:
3030:
3027:
3018:
3014:
3011:
3007:
2998:
2994:
2986:
2983:
2978:
2972:
2967:
2963:
2959:
2956:
2953:
2946:
2934:
2931:
2928:
2924:
2920:
2915:
2911:
2904:
2901:
2896:
2893:
2890:
2886:
2878:
2875:
2870:
2864:
2859:
2855:
2850:
2844:
2839:
2836:
2833:
2829:
2825:
2819:
2816:
2807:
2803:
2800:
2796:
2787:
2783:
2775:
2772:
2767:
2761:
2756:
2752:
2748:
2744:
2740:
2737:
2723:
2720:
2717:
2713:
2709:
2704:
2700:
2693:
2690:
2685:
2682:
2679:
2675:
2667:
2664:
2658:
2653:
2649:
2644:
2639:
2633:
2628:
2625:
2622:
2618:
2614:
2598:
2595:
2582:
2579:
2572:
2563:
2559:
2551:
2548:
2543:
2537:
2532:
2529:
2526:
2522:
2518:
2515:
2508:
2504:
2501:
2487:
2484:
2481:
2477:
2473:
2468:
2464:
2457:
2454:
2449:
2446:
2443:
2439:
2431:
2428:
2422:
2417:
2413:
2408:
2402:
2397:
2394:
2391:
2387:
2378:
2361:
2358:
2351:
2342:
2338:
2330:
2327:
2322:
2316:
2311:
2308:
2305:
2301:
2297:
2294:
2287:
2281:
2268:
2265:
2262:
2258:
2251:
2248:
2240:
2236:
2229:
2223:
2217:
2212:
2209:
2206:
2202:
2150:
2147:
2144:
2138:
2135:
2113:
2110:
2107:
2103:
2099:
2094:
2090:
2086:
2083:
2060:
2057:
2053:
2048:
2039:
2035:
2027:
2024:
2019:
2015:
2011:
1999:
1996:
1993:
1989:
1985:
1980:
1976:
1969:
1966:
1961:
1958:
1955:
1951:
1943:
1940:
1935:
1930:
1906:
1900:
1897:
1894:
1871:
1848:
1842:
1834:
1830:
1826:
1807:
1804:
1801:
1777:
1773:
1770:
1766:
1758:The function
1757:
1743:
1740:
1737:
1723:
1720:
1717:
1713:
1709:
1704:
1700:
1693:
1690:
1685:
1682:
1679:
1675:
1667:
1664:
1658:
1653:
1649:
1645:
1639:
1626:
1623:
1620:
1616:
1609:
1606:
1598:
1594:
1587:
1564:
1561:
1557:
1551:
1548:
1545:
1536:
1533:
1530:
1526:
1522:
1517:
1513:
1506:
1503:
1500:
1475:
1471:
1467:
1462:
1459:
1456:
1452:
1425:
1422:
1419:
1413:
1410:
1385:
1382:
1379:
1375:
1371:
1366:
1362:
1355:
1352:
1347:
1344:
1341:
1337:
1333:
1330:
1310:
1307:
1293:
1290:
1287:
1283:
1279:
1274:
1270:
1263:
1260:
1255:
1252:
1249:
1245:
1237:
1234:
1228:
1223:
1219:
1215:
1209:
1206:
1203:
1194:
1187:
1184:
1176:
1172:
1164:
1161:
1158:
1154:
1149:
1145:
1137:
1134:
1131:
1127:
1120:
1117:
1109:
1105:
1098:
1090:
1087:
1086:
1085:
1082:
1069:
1066:
1063:
1047:
1040:
1037:
1024:
1019:
1015:
1011:
1008:
1001:
995:
982:
979:
976:
972:
965:
962:
954:
950:
943:
937:
931:
926:
923:
920:
916:
904:
896:
881:
878:
875:
871:
864:
861:
853:
849:
842:
830:
825:
822:
819:
815:
803:
795:
789:
783:
775:
761:
758:
755:
752:
749:
746:
743:
740:
737:
734:
712:
709:
706:
702:
698:
693:
689:
685:
680:
676:
673:
670:
664:
661:
638:
632:
629:
626:
623:
620:
616:
609:
606:
603:
597:
594:
591:
588:
583:
579:
569:
544:
541:
538:
534:
527:
524:
516:
512:
505:
493:
488:
485:
482:
478:
466:
458:
452:
446:
438:
421:
418:
415:
404:
388:
380:
376:
358:
342:
339:
336:
330:
327:
317:
307:
305:
301:
297:
293:
289:
284:
270:
260:
256:
242:
233:
231:
226:
225:rectification
220:
218:
216:
213:(cumulative)
210:
206:
202:
198:
194:
193:line segments
190:
186:
182:
178:
170:
169:rectification
165:
156:
154:
150:
145:
129:
113:
110:
107:
101:
98:
90:
86:
81:
79:
75:
71:
70:line segments
66:
64:
60:
53:
49:
45:
40:
32:
19:
13067:Secant cubed
12992:
12985:
12966:Isaac Newton
12936:Brook Taylor
12603:Derivatives
12574:Shell method
12489:
12302:Differential
12095:
12063:
12060:"Arc Length"
12036:
12007:
11987:
11980:
11970:
11963:
11953:
11946:
11924:(2): 89–93.
11921:
11917:
11908:
11875:
11871:
11865:
11853:
11844:
11834:
11809:
11801:
11774:
11770:
11760:
11735:
11727:
11703:Meridian arc
11636:
11589:
11524:
11280:
11147:
11075:
10823:
10805:
10801:
10800: sin(1/
10797:
10793:
10789:
10779:
10704:
10642:
10461:
10457:
10453:
10449:
10445:
10441:
10437:
10435:
10348:
10294:
10290:
10284:
10239:
10226:
10219:
10208:
10198:
10184:
10149:
10146:17th century
10112:
10046:
10029:
9879:
9131:
9126:
9122:
9120:
9111:Circular arc
9100:Simple cases
8915:
8706:
8300:dot products
8186:
7927:
7863:
7609:
7344:
7186:
7109:
7039:
6850:
6623:
6620:
6463:
6285:
6237:
6151:
6128:
6034:
5746:
5695:Curves with
5694:
5600:
5597:
5488:
5325:planar curve
5322:
4918:
4757:
4605:
4487:
4484:
4479:
3785:
3573:
3327:is used. By
3217:
2599:
2596:
2379:
2376:
1083:
776:
570:
439:
319:
287:
285:
234:
224:
221:
212:
195:to create a
174:
168:
146:
82:
77:
73:
67:
58:
57:
51:
43:
13135:of surfaces
12886:and numbers
12848:Dirichlet's
12818:Telescoping
12771:Alternating
12359:L'HĂ´pital's
12156:Precalculus
12078:Ed Pegg Jr.
11647:proper time
11643:world lines
10936:defined by
10916:a curve in
10164:John Wallis
9655:right-angle
4480:rectifiable
4109:approaches
292:orientation
257:rectifiable
171:of a curve.
42:Arc length
13187:Categories
12931:Adequality
12617:Divergence
12490:Arc length
12287:Derivative
12074:Arc Length
11860:, p. F-254
11719:References
10786:Koch curve
10713:See also:
10119:Archimedes
9964:arcminutes
9870:See also:
9834: turn
9737: grad
9689: grad
9406:semicircle
8050:-axis and
6459:chain rule
5733:hyperbolic
5309:See also:
4606:where the
1884:such that
314:See also:
261:arc length
187:number of
59:Arc length
13130:of curves
13125:Curvature
13012:Integrals
12806:Maclaurin
12786:Geometric
12677:Geometric
12627:Laplacian
12339:linearity
12179:Factorial
12065:MathWorld
12043:EMS Press
11900:123757069
11892:0746-8342
11793:0022-247X
11713:Sinuosity
11683:Geodesics
11645:) is the
11598:γ
11561:γ
11553:∈
11537:γ
11478:γ
11433:γ
11396:γ
11348:∑
11344:±
11328:∫
11318:γ
11312:ℓ
11248:γ
11220:γ
11195:∫
11185:γ
11179:ℓ
11156:γ
11114:γ
11089:γ
11044:∈
11016:γ
11009:…
10988:γ
10969:γ
10901:→
10880:γ
10796:) =
10748:
10742:⋅
10663:ε
10589:ε
10569:ε
10543:ε
10397:−
10182:in 1691.
10109:Antiquity
10010:θ
9950:θ
9906:θ
9822:θ
9793:π
9766:θ
9728:θ
9680:θ
9674:π
9637:θ
9608:∘
9599:θ
9560:∘
9551:θ
9545:π
9512:θ
9486:θ
9453:θ
9422:π
9386:π
9359:π
9327:π
9236:θ
9014:θ
8925:∫
8869:ϕ
8855:θ
8852:
8805:θ
8716:∫
8676:ϕ
8667:θ
8664:
8624:θ
8562:ϕ
8546:ϕ
8536:⋅
8531:ϕ
8499:θ
8483:θ
8473:⋅
8468:θ
8410:⋅
8321:⋅
8279:ϕ
8273:ϕ
8256:θ
8250:θ
8206:∘
8167:θ
8164:
8152:ϕ
8149:
8143:θ
8140:
8128:ϕ
8125:
8119:θ
8116:
8098:ϕ
8092:θ
8058:ϕ
8018:θ
7986:ϕ
7971:θ
7913:θ
7884:θ
7847:θ
7812:θ
7770:θ
7749:θ
7745:∫
7708:θ
7619:∫
7579:θ
7517:θ
7501:θ
7491:⋅
7486:θ
7464:θ
7445:θ
7435:⋅
7369:⋅
7323:θ
7317:θ
7273:∘
7211:∘
7167:θ
7164:
7152:θ
7149:
7131:θ
7083:θ
6684:⋅
6482:∘
6414:∘
6219:π
6177:−
6162:
6098:−
6047:−
6043:∫
6002:−
5913:−
5895:−
5817:−
5809:∈
5771:−
5615:∫
5204:∫
5185:φ
5154:φ
5131:φ
5099:∫
5056:φ
5040:φ
5008:∫
4950:∫
4869:→
4843:−
4839:φ
4835:∘
4816:bijection
4787:→
4766:φ
4664:−
4650:⋯
4577:−
4560:−
4514:∑
4422:→
4393:function
4291:Δ
4237:∑
4227:Δ
4214:Δ
4201:−
4184:−
4138:∑
4074:→
4071:ε
4048:→
4042:ε
4036:δ
4014:∞
4011:→
3985:ε
3979:δ
3965:−
3927:−
3918:ε
3909:Δ
3903:ε
3883:θ
3832:∫
3769:Δ
3763:ε
3718:−
3710:θ
3690:−
3679:−
3663:θ
3652:−
3619:∫
3588:∑
3581:Δ
3556:ε
3550:δ
3541:Δ
3518:ε
3512:δ
3498:−
3466:ε
3421:−
3402:−
3391:−
3375:θ
3364:−
3315:θ
3264:∫
3200:θ
3157:−
3138:−
3127:−
3111:θ
3100:−
3062:∫
3041:∑
3034:Δ
3015:θ
2964:∫
2960:−
2957:θ
2932:−
2921:−
2905:θ
2894:−
2856:∫
2830:∑
2823:Δ
2820:≦
2804:θ
2753:∫
2749:−
2741:θ
2721:−
2710:−
2694:θ
2683:−
2650:∫
2619:∑
2612:Δ
2577:Δ
2523:∑
2519:−
2513:Δ
2505:θ
2485:−
2474:−
2458:θ
2447:−
2414:∫
2388:∑
2356:Δ
2302:∑
2298:−
2292:Δ
2279:Δ
2266:−
2249:−
2203:∑
2181:∞
2178:→
2139:∈
2136:θ
2111:−
2100:−
2081:Δ
2061:ε
2016:−
1997:−
1986:−
1970:θ
1959:−
1907:ε
1901:δ
1892:Δ
1872:ε
1849:ε
1843:δ
1835:function
1741:θ
1721:−
1710:−
1694:θ
1683:−
1650:∫
1637:Δ
1624:−
1607:−
1565:θ
1555:Δ
1549:θ
1534:−
1523:−
1460:−
1414:∈
1411:θ
1383:−
1372:−
1356:θ
1345:−
1311:θ
1291:−
1280:−
1264:θ
1253:−
1220:∫
1213:Δ
1162:−
1150:∫
1135:−
1118:−
1016:∫
1006:Δ
993:Δ
980:−
963:−
917:∑
911:∞
908:→
879:−
862:−
816:∑
810:∞
807:→
753:…
710:−
699:−
674:−
659:Δ
636:Δ
607:−
542:−
525:−
479:∑
473:∞
470:→
375:injective
349:→
331::
209:summation
120:→
102::
85:injective
13120:Manifold
12853:Integral
12796:Infinite
12791:Harmonic
12776:Binomial
12622:Gradient
12565:Volumes
12376:Quotient
12317:Notation
12148:Calculus
11845:nist.gov
11657:See also
11540:′
11223:′
10223:parabola
10176:catenary
10135:polygons
10127:calculus
10024:gradians
9884:and the
9657:), then
9182:diameter
8679:′
8627:′
8591:′
8565:′
8502:′
8439:′
8282:′
8259:′
8236:′
7928:Now let
7582:′
7546:′
7520:′
7467:′
7459:′
7398:′
7326:′
7303:′
7224:′
6950:′
6927:′
6826:′
6800:′
6792:′
6758:′
6729:′
6706:′
6675:′
6652:′
6603:′
6580:′
6549:′
6541:′
6427:′
5729:elliptic
5717:parabola
5701:catenary
5229:′
5157:′
5124:′
5059:′
5033:′
4975:′
4608:supremum
4327:′
4266:′
3855:′
3803:′
3730:′
3637:′
3533:so that
3433:′
3349:′
3287:′
3235:′
3169:′
3085:′
2987:′
2879:′
2776:′
2668:′
2552:′
2432:′
2331:′
2028:′
1944:′
1922:implies
1774:′
1668:′
1441:maps to
1238:′
1188:′
1041:′
259:and the
217:distance
76:. For a
13057:inverse
13045:inverse
12971:Fluxion
12781:Fourier
12647:Stokes'
12642:Green's
12364:Product
12224:Tangent
12084:, 2007.
12045:, 2001
12000:Sources
11938:2308256
11525:where
10287:tangent
10168:cycloid
9971:⁄
9528:, then
9526:degrees
9467:radians
9408:, then
9204:is its
9180:is its
9127:spatium
6966:(where
6881:is the
6851:(where
5709:cycloid
4818:. Then
215:chordal
179:in the
13203:Length
13198:Curves
13140:Tensor
13062:Secant
12828:Abel's
12811:Taylor
12702:Matrix
12652:Gauss'
12234:Limits
12214:Secant
12204:Radian
12014:
11936:
11898:
11890:
11822:
11791:
11748:
10297:had a
10285:whose
10193:, the
10154:: the
10022:is in
9977:degree
9962:is in
9778:is in
9649:is in
9524:is in
9465:is in
9250:centre
9158:circle
9154:radius
8345:where
6511:
6159:arcsin
5705:circle
5394:where
5249:
5170:
5079:
4995:
3945:, and
3704:
3194:
2951:
2735:
2499:
2073:where
1735:
1323:where
1305:
1201:
1091:shows
1061:
571:where
373:be an
296:origin
201:length
189:points
185:finite
13004:Lists
12863:Ratio
12801:Power
12537:Euler
12354:Chain
12344:Power
12219:Slope
11934:JSTOR
11896:S2CID
10844:be a
10299:slope
10129:, by
10050:0.539
9979:), or
9886:metre
9780:turns
9651:grads
9469:then
9156:of a
6252:1.570
6135:1.570
5323:If a
3786:with
1403:over
651:with
403:limit
181:plane
177:curve
63:curve
46:of a
12873:Term
12868:Root
12607:Curl
12012:ISBN
11888:ISSN
11820:ISBN
11789:ISSN
11746:ISBN
11076:and
10824:Let
10812:and
10213:and
9874:and
9287:and
8365:and
8298:All
7040:Let
6999:and
6286:Let
5937:and
5731:and
5723:and
5482:The
5444:and
4672:<
4653:<
4647:<
4634:<
4097:<
3956:>
3760:<
3547:<
3489:>
3478:for
3463:<
2128:and
2058:<
1898:<
1493:and
727:for
377:and
320:Let
302:and
87:and
12349:Sum
12076:by
11926:doi
11880:doi
11816:137
11779:doi
11775:445
11637:In
11610:at
10848:,
10745:sin
10456:to
10301:of
10289:at
10178:by
10170:by
10158:by
10052:956
10041:600
10034:000
9982:if
9922:if
9758:If
9733:400
9685:200
9629:If
9604:360
9556:180
9504:If
9445:If
8843:sin
8655:sin
8161:cos
8146:sin
8137:sin
8122:cos
8113:sin
7161:sin
7146:cos
7032:).
6268:1.7
6263:727
6260:794
6257:326
6254:796
6240:1.3
6238:by
6146:177
6143:808
6140:326
6137:796
5414:is
5327:in
4700:of
4510:sup
4446:on
4354:on
4310:of
4063:so
901:lim
800:lim
463:lim
306:).
13189::
12080:,
12062:.
12041:,
12035:,
11932:.
11922:60
11920:.
11894:.
11886:.
11876:46
11874:.
11843:.
11818:.
11787:.
11773:.
11769:.
11744:.
11742:51
11301:,
10872:,
10468::
10462:AC
10450:AC
10444:+
10293:=
10217:.
10201:.
10142:.
10055:80
10039:21
10032:40
9973:60
9208:,
9184:,
9160:,
9129:.
7925:.
6812:22
6783:12
6744:11
6272:10
6244:10
5739:.
5719:,
5715:,
5711:,
5707:,
5703:,
4482:.
3895:,
286:A
283:.
232:.
219:.
175:A
65:.
12140:e
12133:t
12126:v
12107:.
12068:.
12020:.
11940:.
11928::
11902:.
11882::
11847:.
11828:.
11795:.
11781::
11754:.
11621:.
11618:t
11575:M
11570:)
11567:t
11564:(
11557:T
11550:)
11547:t
11544:(
11521:,
11509:t
11506:d
11498:t
11495:d
11490:)
11487:)
11484:t
11481:(
11475:(
11470:j
11466:x
11462:d
11453:t
11450:d
11445:)
11442:)
11439:t
11436:(
11430:(
11425:i
11421:x
11417:d
11411:)
11408:)
11405:)
11402:t
11399:(
11393:(
11390:x
11387:(
11382:j
11379:i
11375:g
11369:n
11364:1
11361:=
11358:j
11355:,
11352:i
11337:1
11332:0
11324:=
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11315:(
11289:x
11277:,
11265:t
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11237:|
11233:)
11230:t
11227:(
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11210:|
11204:1
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11191:=
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11031:)
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10516:+
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10026:.
9990:s
9969:1
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9903:=
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9847:.
9830:1
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9665:s
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8217:=
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8210:C
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8110:r
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7507:)
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7440:x
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7330:.
7312:x
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7300:r
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7289:x
7284:=
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7277:C
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7262:D
7242:.
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7234:)
7231:t
7228:(
7220:)
7215:C
7207:x
7202:(
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7095:)
7092:)
7089:t
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7080:,
7077:)
7074:t
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7062:=
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6984:=
6979:1
6975:u
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6933:(
6923:)
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6797:v
6789:u
6779:g
6775:2
6772:+
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6762:)
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6736:=
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6726:v
6720:v
6715:x
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6687:(
6680:)
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6661:x
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6638:x
6632:(
6607:.
6600:v
6594:v
6589:x
6584:+
6577:u
6571:u
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6561:=
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6546:v
6538:u
6532:(
6526:)
6521:v
6516:x
6506:u
6501:x
6496:(
6493:=
6490:)
6486:C
6478:x
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6471:D
6445:.
6441:|
6437:)
6434:t
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6418:C
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6399:|
6378:)
6375:)
6372:t
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6366:v
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6360:)
6357:t
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6351:u
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6345:=
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6339:t
6336:(
6332:C
6311:)
6308:v
6305:,
6302:u
6299:(
6295:x
6270:Ă—
6242:Ă—
6222:2
6214:=
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6205:/
6199:2
6192:2
6188:/
6182:2
6171:|
6165:x
6115:.
6106:2
6102:x
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6058:/
6052:2
6020:,
6016:)
6010:2
6006:x
5999:1
5995:(
5989:/
5984:1
5981:=
5976:2
5972:)
5968:x
5965:d
5961:/
5957:y
5954:d
5951:(
5948:+
5945:1
5921:2
5917:x
5910:1
5903:/
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5892:=
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5886:d
5882:/
5878:y
5875:d
5854:]
5850:2
5846:/
5840:2
5835:,
5832:2
5828:/
5822:2
5813:[
5806:x
5786:.
5779:2
5775:x
5768:1
5763:=
5760:y
5681:.
5678:x
5675:d
5667:2
5662:)
5656:x
5653:d
5648:y
5645:d
5639:(
5634:+
5631:1
5624:b
5619:a
5611:=
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5584:.
5581:x
5578:d
5570:2
5565:)
5559:x
5556:d
5551:y
5548:d
5542:(
5537:+
5534:1
5529:=
5522:2
5518:y
5514:d
5511:+
5506:2
5502:x
5498:d
5470:.
5467:)
5464:t
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5455:=
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5432:t
5429:=
5426:x
5402:f
5382:,
5379:)
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5364:y
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5280:g
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5239:)
5236:u
5233:(
5226:g
5220:|
5213:d
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5200:=
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5164:t
5161:(
5148:|
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5121:g
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5108:b
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5095:=
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5082:d
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5069:)
5066:t
5063:(
5052:)
5049:)
5046:t
5043:(
5037:(
5030:g
5024:|
5017:b
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5004:=
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4985:)
4982:t
4979:(
4972:f
4966:|
4959:b
4954:a
4946:=
4939:)
4936:f
4933:(
4930:L
4904:.
4901:f
4879:n
4874:R
4866:]
4863:d
4860:,
4857:c
4854:[
4851::
4846:1
4832:f
4829:=
4826:g
4802:]
4799:d
4796:,
4793:c
4790:[
4784:]
4781:b
4778:,
4775:a
4772:[
4769::
4743:f
4723:.
4720:]
4717:b
4714:,
4711:a
4708:[
4688:b
4685:=
4680:N
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4667:1
4661:N
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4642:1
4638:t
4629:0
4625:t
4621:=
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4590:|
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4580:1
4574:i
4570:t
4566:(
4563:f
4557:)
4552:i
4548:t
4544:(
4541:f
4536:|
4529:N
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4521:=
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4507:=
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4501:f
4498:(
4495:L
4466:]
4463:b
4460:,
4457:a
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4432:n
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4416:b
4413:,
4410:a
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4401:f
4377:.
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4371:b
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4337:)
4334:t
4331:(
4324:f
4319:|
4294:t
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4283:)
4278:i
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4270:(
4263:f
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4252:N
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4244:=
4241:i
4233:=
4230:t
4223:|
4217:t
4209:)
4204:1
4198:i
4194:t
4190:(
4187:f
4181:)
4176:i
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4165:f
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4153:N
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4145:=
4142:i
4117:0
4077:0
4051:0
4045:)
4039:(
4017:,
4008:N
3988:)
3982:(
3975:/
3971:)
3968:a
3962:b
3959:(
3953:N
3933:)
3930:a
3924:b
3921:(
3915:=
3912:t
3906:N
3880:d
3876:|
3872:)
3867:i
3863:t
3859:(
3852:f
3847:|
3841:1
3836:0
3828:=
3824:|
3820:)
3815:i
3811:t
3807:(
3800:f
3795:|
3772:t
3766:N
3756:)
3751:|
3747:)
3742:i
3738:t
3734:(
3727:f
3722:|
3714:|
3707:d
3701:)
3698:)
3693:1
3687:i
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3666:(
3660:+
3655:1
3649:i
3645:t
3641:(
3634:f
3628:1
3623:0
3614:|
3609:(
3603:N
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3595:=
3592:i
3584:t
3559:)
3553:(
3544:t
3521:)
3515:(
3508:/
3504:)
3501:a
3495:b
3492:(
3486:N
3459:|
3454:|
3450:)
3445:i
3441:t
3437:(
3430:f
3425:|
3417:|
3413:)
3410:)
3405:1
3399:i
3395:t
3386:i
3382:t
3378:(
3372:+
3367:1
3361:i
3357:t
3353:(
3346:f
3341:|
3336:|
3312:d
3308:|
3304:)
3299:i
3295:t
3291:(
3284:f
3279:|
3273:1
3268:0
3260:=
3256:|
3252:)
3247:i
3243:t
3239:(
3232:f
3227:|
3197:d
3190:|
3186:)
3181:i
3177:t
3173:(
3166:f
3161:|
3153:|
3149:)
3146:)
3141:1
3135:i
3131:t
3122:i
3118:t
3114:(
3108:+
3103:1
3097:i
3093:t
3089:(
3082:f
3077:|
3071:1
3066:0
3056:N
3051:1
3048:=
3045:i
3037:t
3031:=
3019:)
3012:d
3008:|
3004:)
2999:i
2995:t
2991:(
2984:f
2979:|
2973:1
2968:0
2954:d
2947:|
2943:)
2940:)
2935:1
2929:i
2925:t
2916:i
2912:t
2908:(
2902:+
2897:1
2891:i
2887:t
2883:(
2876:f
2871:|
2865:1
2860:0
2851:(
2845:N
2840:1
2837:=
2834:i
2826:t
2808:)
2801:d
2797:|
2793:)
2788:i
2784:t
2780:(
2773:f
2768:|
2762:1
2757:0
2745:|
2738:d
2732:)
2729:)
2724:1
2718:i
2714:t
2705:i
2701:t
2697:(
2691:+
2686:1
2680:i
2676:t
2672:(
2665:f
2659:1
2654:0
2645:|
2640:(
2634:N
2629:1
2626:=
2623:i
2615:t
2583:.
2580:t
2573:|
2569:)
2564:i
2560:t
2556:(
2549:f
2544:|
2538:N
2533:1
2530:=
2527:i
2516:t
2509:|
2502:d
2496:)
2493:)
2488:1
2482:i
2478:t
2469:i
2465:t
2461:(
2455:+
2450:1
2444:i
2440:t
2436:(
2429:f
2423:1
2418:0
2409:|
2403:N
2398:1
2395:=
2392:i
2362:.
2359:t
2352:|
2348:)
2343:i
2339:t
2335:(
2328:f
2323:|
2317:N
2312:1
2309:=
2306:i
2295:t
2288:|
2282:t
2274:)
2269:1
2263:i
2259:t
2255:(
2252:f
2246:)
2241:i
2237:t
2233:(
2230:f
2224:|
2218:N
2213:1
2210:=
2207:i
2175:N
2154:]
2151:1
2148:,
2145:0
2142:[
2114:1
2108:i
2104:t
2095:i
2091:t
2087:=
2084:t
2054:|
2049:|
2045:)
2040:i
2036:t
2032:(
2025:f
2020:|
2012:|
2008:)
2005:)
2000:1
1994:i
1990:t
1981:i
1977:t
1973:(
1967:+
1962:1
1956:i
1952:t
1948:(
1941:f
1936:|
1931:|
1910:)
1904:(
1895:t
1852:)
1846:(
1811:]
1808:b
1805:,
1802:a
1799:[
1778:|
1771:f
1767:|
1744:.
1738:d
1732:)
1729:)
1724:1
1718:i
1714:t
1705:i
1701:t
1697:(
1691:+
1686:1
1680:i
1676:t
1672:(
1665:f
1659:1
1654:0
1646:=
1640:t
1632:)
1627:1
1621:i
1617:t
1613:(
1610:f
1604:)
1599:i
1595:t
1591:(
1588:f
1562:d
1558:t
1552:=
1546:d
1542:)
1537:1
1531:i
1527:t
1518:i
1514:t
1510:(
1507:=
1504:t
1501:d
1481:]
1476:i
1472:t
1468:,
1463:1
1457:i
1453:t
1449:[
1429:]
1426:1
1423:,
1420:0
1417:[
1391:)
1386:1
1380:i
1376:t
1367:i
1363:t
1359:(
1353:+
1348:1
1342:i
1338:t
1334:=
1331:t
1308:d
1302:)
1299:)
1294:1
1288:i
1284:t
1275:i
1271:t
1267:(
1261:+
1256:1
1250:i
1246:t
1242:(
1235:f
1229:1
1224:0
1216:t
1210:=
1207:t
1204:d
1198:)
1195:t
1192:(
1185:f
1177:i
1173:t
1165:1
1159:i
1155:t
1146:=
1143:)
1138:1
1132:i
1128:t
1124:(
1121:f
1115:)
1110:i
1106:t
1102:(
1099:f
1070:.
1067:t
1064:d
1056:|
1051:)
1048:t
1045:(
1038:f
1032:|
1025:b
1020:a
1012:=
1009:t
1002:|
996:t
988:)
983:1
977:i
973:t
969:(
966:f
960:)
955:i
951:t
947:(
944:f
938:|
932:N
927:1
924:=
921:i
905:N
897:=
892:|
887:)
882:1
876:i
872:t
868:(
865:f
859:)
854:i
850:t
846:(
843:f
838:|
831:N
826:1
823:=
820:i
804:N
796:=
793:)
790:f
787:(
784:L
762:.
759:N
756:,
750:,
747:1
744:,
741:0
738:=
735:i
713:1
707:i
703:t
694:i
690:t
686:=
681:N
677:a
671:b
665:=
662:t
639:t
633:i
630:+
627:a
624:=
621:N
617:/
613:)
610:a
604:b
601:(
598:i
595:+
592:a
589:=
584:i
580:t
555:|
550:)
545:1
539:i
535:t
531:(
528:f
522:)
517:i
513:t
509:(
506:f
501:|
494:N
489:1
486:=
483:i
467:N
459:=
456:)
453:f
450:(
447:L
425:]
422:b
419:,
416:a
413:[
389:f
359:n
354:R
346:]
343:b
340:,
337:a
334:[
328:f
271:L
243:L
130:n
125:R
117:]
114:b
111:,
108:a
105:[
99:f
54:.
52:θ
44:s
20:)
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