4105:
3287:
4100:{\displaystyle {\begin{aligned}A_{x}&=\int _{a}^{b}\int _{0}^{2\pi }\left\|{\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}\right\|\ d\theta \ dt\\&=\int _{a}^{b}\int _{0}^{2\pi }\left\|y\left\langle y\cos(\theta ){\frac {dx}{dt}},y\sin(\theta ){\frac {dx}{dt}},y{\frac {dy}{dt}}\right\rangle \right\|\ d\theta \ dt\\&=\int _{a}^{b}\int _{0}^{2\pi }y{\sqrt {\cos ^{2}(\theta )\left({\frac {dx}{dt}}\right)^{2}+\sin ^{2}(\theta )\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ d\theta \ dt\\&=\int _{a}^{b}\int _{0}^{2\pi }y{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ d\theta \ dt\\&=\int _{a}^{b}2\pi y{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ dt\end{aligned}}}
1647:
1629:
1617:
4542:
31:
2662:
3212:
1004:
229:
4813:
4220:
2386:
2173:
43:
2923:
988:
4557:
1889:
4537:{\displaystyle {\begin{aligned}V_{x}&=\int _{\alpha }^{\beta }\left(\pi r^{2}\sin ^{2}{\theta }\cos {\theta }\,{\frac {dr}{d\theta }}-\pi r^{3}\sin ^{3}{\theta }\right)d\theta \,,\\V_{y}&=\int _{\alpha }^{\beta }\left(\pi r^{2}\sin {\theta }\cos ^{2}{\theta }\,{\frac {dr}{d\theta }}+\pi r^{3}\cos ^{3}{\theta }\right)d\theta \,.\end{aligned}}}
2657:{\displaystyle A_{x}=\iint _{S}dS=\iint _{\times }\left\|{\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}\right\|\ d\theta \ dt=\int _{a}^{b}\int _{0}^{2\pi }\left\|{\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}\right\|\ d\theta \ dt.}
628:
3207:{\displaystyle {\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}=\left\langle y\cos(\theta ){\frac {dx}{dt}},y\sin(\theta ){\frac {dx}{dt}},y{\frac {dy}{dt}}\right\rangle =y\left\langle \cos(\theta ){\frac {dx}{dt}},\sin(\theta ){\frac {dx}{dt}},{\frac {dy}{dt}}\right\rangle }
1602:
2815:
1874:
46:
4808:{\displaystyle {\begin{aligned}A_{x}&=\int _{\alpha }^{\beta }2\pi r\sin {\theta }\,{\sqrt {r^{2}+\left({\frac {dr}{d\theta }}\right)^{2}}}\,d\theta \,,\\A_{y}&=\int _{\alpha }^{\beta }2\pi r\cos {\theta }\,{\sqrt {r^{2}+\left({\frac {dr}{d\theta }}\right)^{2}}}\,d\theta \,,\end{aligned}}}
45:
50:
49:
44:
51:
2168:{\displaystyle {\begin{aligned}A_{x}&=\int _{a}^{b}2\pi y\,{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\,dt\,,\\A_{y}&=\int _{a}^{b}2\pi x\,{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\,dt\,.\end{aligned}}}
1331:
2914:
48:
2669:
1696:
983:{\displaystyle V=\iiint _{D}dV=\int _{a}^{b}\int _{g(z)}^{f(z)}\int _{0}^{2\pi }r\,d\theta \,dr\,dz=2\pi \int _{a}^{b}\int _{g(z)}^{f(z)}r\,dr\,dz=2\pi \int _{a}^{b}{\frac {1}{2}}r^{2}\Vert _{g(z)}^{f(z)}\,dz=\pi \int _{a}^{b}(f(z)^{2}-g(z)^{2})\,dz}
2343:
3292:
200:. To apply these methods, it is easiest to draw the graph in question; identify the area that is to be revolved about the axis of revolution; determine the volume of either a disc-shaped slice of the solid, with thickness
400:
2819:
47:
3282:
1161:
4562:
4225:
1894:
1701:
1256:
486:
4180:
2225:
1597:{\displaystyle V=\iiint _{D}dV=\int _{a}^{b}\int _{g(r)}^{f(r)}\int _{0}^{2\pi }r\,d\theta \,dz\,dr=2\pi \int _{a}^{b}\int _{g(r)}^{f(r)}r\,dz\,dr=2\pi \int _{a}^{b}r(f(r)-g(r))\,dr.}
2378:
4215:
4148:
2230:
2810:{\displaystyle {\frac {\partial \mathbf {r} }{\partial t}}=\left\langle {\frac {dy}{dt}}\cos(\theta ),{\frac {dy}{dt}}\sin(\theta ),{\frac {dx}{dt}}\right\rangle ,}
1869:{\displaystyle {\begin{aligned}V_{x}&=\int _{a}^{b}\pi y^{2}\,{\frac {dx}{dt}}\,dt\,,\\V_{y}&=\int _{a}^{b}\pi x^{2}\,{\frac {dy}{dt}}\,dt\,.\end{aligned}}}
4921:
302:
212:
approaches 0, a value which may be found by evaluating a suitable integral. A more rigorous justification can be given by attempting to evaluate a
1071:
625:
and the multivariate change of variables formula, the disk method may be derived in a straightforward manner by (denoting the solid as D):
1181:
420:
4107:
where the same trigonometric identity was used again. The derivation for a surface obtained by revolving around the y-axis is similar.
4957:
3217:
2909:{\displaystyle {\frac {\partial \mathbf {r} }{\partial \theta }}=\left\langle -y\sin(\theta ),y\cos(\theta ),0\right\rangle }
17:
4901:
4874:
4929:
4153:
1878:
Under the same circumstances the areas of the surfaces of the solids generated by revolving the curve around the
2227:
then its corresponding surface of revolution when revolved around the x-axis has
Cartesian coordinates given by
2180:
2348:
1328:
This method may be derived with the same triple integral, this time with a different order of integration:
1011:
The shell method (sometimes referred to as the "cylinder method") is used when the slice that was drawn is
135:
5006:
4185:
4118:
1646:
4891:
217:
94:
2338:{\displaystyle \mathbf {r} (t,\theta )=\langle y(t)\cos(\theta ),y(t)\sin(\theta ),x(t)\rangle }
4966:
4864:
4838:
4217:, the volumes of the solids generated by revolving the curve around the x-axis or y-axis are
107:
102:
56:
35:
4949:
8:
1685:
in some interval , the volumes of the solids generated by revolving the curve around the
1662:
1650:
622:
168:
4982:
89:
2177:
This can also be derived from multivariable integration. If a plane curve is given by
1628:
1616:
4979:
4953:
4941:
4897:
4870:
4823:
998:
197:
4546:
The areas of the surfaces of the solids generated by revolving the curve around the
4828:
2381:
1325:). Summing up all of the surface areas along the interval gives the total volume.
237:
193:
5011:
213:
1287:-axis; it forms a cylindrical shell. The lateral surface area of a cylinder is
4937:
149:
72:
5000:
2917:
1654:
84:
30:
4971:
4833:
490:
The method can be visualized by considering a thin horizontal rectangle at
192:
Two common methods for finding the volume of a solid of revolution are the
153:
80:
1022:
The volume of the solid formed by rotating the area between the curves of
253:
The volume of the solid formed by rotating the area between the curves of
1260:
The method can be visualized by considering a thin vertical rectangle at
395:{\displaystyle V=\pi \int _{a}^{b}\left|f(y)^{2}-g(y)^{2}\right|\,dy\,.}
98:
4987:
1634:
The shapes in motion, showing the solids of revolution formed by each
146:
1003:
610:. The limit of the Riemann sum of the volumes of the discs between
228:
127:
76:
64:
152:
of a solid of revolution. The element is created by rotating a
4843:
171:
157:
123:
119:
115:
114:
Assuming that the curve does not cross the axis, the solid's
105:
created by this revolution and which bounds the solid is the
4977:
131:
242:
The disc method is used when the slice that was drawn is
3277:{\displaystyle \sin ^{2}(\theta )+\cos ^{2}(\theta )=1}
1156:{\displaystyle V=2\pi \int _{a}^{b}x|f(x)-g(x)|\,dx\,.}
595:). The volume of each infinitesimal disc is therefore
4560:
4223:
4188:
4156:
4121:
3290:
3220:
2926:
2822:
2672:
2389:
2351:
2233:
2183:
1892:
1699:
1334:
1184:
1074:
631:
423:
305:
208:; and then find the limiting sum of these volumes as
4807:
4536:
4209:
4174:
4142:
4099:
3276:
3206:
2908:
2809:
2656:
2372:
2337:
2219:
2167:
1868:
1596:
1251:{\displaystyle V=2\pi \int _{a}^{b}x|f(x)|\,dx\,.}
1250:
1155:
982:
481:{\displaystyle V=\pi \int _{a}^{b}f(y)^{2}\,dy\,.}
480:
413:(e.g. revolving an area between the curve and the
394:
520:-axis; it forms a ring (or disc in the case that
4998:
4896:(6th ed.). Tata McGraw-Hill. p. 6.90.
4952:. McGraw-Hill Professional. pp. 244–248.
4936:
4175:{\displaystyle \alpha \leq \theta \leq \beta }
1653:: study of a vase as a solid of revolution by
1015:the axis of revolution; i.e. when integrating
992:
246:the axis of revolution; i.e. when integrating
2332:
2257:
2214:
2184:
866:
4869:. Discovery Publishing House. p. 168.
3284:was used. With this cross product, we get
1174:(e.g. revolving an area between curve and
516:on the bottom, and revolving it about the
220:with two different orders of integration.
34:Rotating a curve. The surface formed is a
4797:
4790:
4737:
4677:
4670:
4617:
4526:
4460:
4373:
4307:
2666:Computing the partial derivatives yields
2220:{\displaystyle \langle x(t),y(t)\rangle }
2157:
2150:
2072:
2023:
2016:
1938:
1858:
1851:
1830:
1777:
1770:
1749:
1584:
1517:
1510:
1443:
1436:
1429:
1244:
1237:
1149:
1142:
973:
898:
814:
807:
740:
733:
726:
474:
467:
388:
381:
2380:. Then the surface area is given by the
1645:
1002:
227:
41:
29:
2373:{\displaystyle 0\leq \theta \leq 2\pi }
14:
4999:
4862:
4978:
4889:
187:
38:; it encloses a solid of revolution.
24:
3388:
3378:
3363:
3353:
2965:
2955:
2940:
2930:
2836:
2826:
2686:
2676:
2619:
2609:
2594:
2584:
2508:
2498:
2483:
2473:
1641:
584:is the inner radius (in this case
569:is the outer radius (in this case
204:, or a cylindrical shell of width
25:
5023:
4922:"Volumes of Solids of Revolution"
3214:where the trigonometric identity
1609:Solid of revolution demonstration
232:Disc integration about the y-axis
4866:Application Of Integral Calculus
4210:{\displaystyle f(\theta )\geq 0}
3382:
3357:
2959:
2934:
2830:
2680:
2613:
2588:
2502:
2477:
2235:
1627:
1615:
136:Pappus's second centroid theorem
1661:When a curve is defined by its
27:Type of three-dimensional shape
4883:
4856:
4198:
4192:
4137:
4131:
3736:
3730:
3676:
3670:
3587:
3532:
3526:
3491:
3485:
3464:
3398:
3346:
3265:
3259:
3240:
3234:
3153:
3147:
3115:
3109:
3038:
3032:
2997:
2991:
2892:
2886:
2871:
2865:
2773:
2767:
2735:
2729:
2629:
2577:
2518:
2466:
2460:
2445:
2439:
2427:
2329:
2323:
2314:
2308:
2299:
2293:
2284:
2278:
2269:
2263:
2251:
2239:
2211:
2205:
2196:
2190:
1581:
1578:
1572:
1563:
1557:
1551:
1502:
1496:
1488:
1482:
1403:
1397:
1389:
1383:
1233:
1229:
1223:
1216:
1138:
1134:
1128:
1119:
1113:
1106:
970:
961:
954:
939:
932:
926:
893:
887:
879:
873:
799:
793:
785:
779:
700:
694:
686:
680:
621:Assuming the applicability of
458:
451:
367:
360:
345:
338:
223:
101:(except at its boundary). The
13:
1:
4928:. 12 Apr 2011. Archived from
4914:
4110:
1283:, and revolving it about the
4143:{\displaystyle r=f(\theta )}
1306:is the height (in this case
1298:is the radius (in this case
163:) around some axis (located
7:
4863:Sharma, A. K. (2005).
4817:
993:Shell Method of Integration
198:shell method of integration
130:multiplied by the figure's
10:
5028:
996:
235:
126:described by the figure's
4890:Singh, Ravish R. (1993).
1178:-axis), this reduces to:
553:. The area of a ring is
417:-axis), this reduces to:
4849:
1019:the axis of revolution.
250:the axis of revolution.
4893:Engineering Mathematics
218:cylindrical coordinates
167:units away), so that a
4809:
4538:
4211:
4176:
4144:
4101:
3278:
3208:
2910:
2811:
2658:
2374:
2339:
2221:
2169:
1870:
1658:
1598:
1252:
1157:
1008:
984:
618:becomes integral (1).
482:
396:
233:
60:
55:Solids of revolution (
39:
4983:"Solid of Revolution"
4839:Surface of revolution
4810:
4539:
4212:
4177:
4145:
4102:
3279:
3209:
2911:
2812:
2659:
2375:
2340:
2222:
2170:
1871:
1649:
1599:
1253:
1158:
1006:
985:
531:), with outer radius
483:
397:
231:
108:surface of revolution
54:
36:surface of revolution
33:
4558:
4221:
4186:
4154:
4119:
3288:
3218:
2924:
2820:
2670:
2387:
2349:
2231:
2181:
1890:
1697:
1332:
1182:
1072:
629:
421:
303:
4716:
4596:
4412:
4259:
3997:
3874:
3856:
3651:
3633:
3462:
3444:
3344:
3326:
2575:
2557:
2062:
1928:
1886:-axis are given by
1816:
1735:
1693:-axis are given by
1651:Mathematics and art
1547:
1506:
1473:
1425:
1407:
1374:
1211:
1101:
925:
897:
844:
803:
770:
722:
704:
671:
447:
329:
184:units is enclosed.
143:representative disc
69:solid of revolution
4980:Weisstein, Eric W.
4942:Mendelson, Elliott
4805:
4803:
4702:
4582:
4534:
4532:
4398:
4245:
4207:
4172:
4140:
4115:For a polar curve
4097:
4095:
3983:
3857:
3842:
3634:
3619:
3445:
3430:
3327:
3312:
3274:
3204:
2916:and computing the
2906:
2807:
2654:
2558:
2543:
2370:
2335:
2217:
2165:
2163:
2048:
1914:
1866:
1864:
1802:
1721:
1659:
1622:The shapes at rest
1594:
1533:
1474:
1459:
1408:
1375:
1360:
1248:
1197:
1153:
1087:
1068:-axis is given by
1009:
980:
911:
865:
830:
771:
756:
705:
672:
657:
478:
433:
392:
315:
299:-axis is given by
234:
188:Finding the volume
90:axis of revolution
61:
57:Matemateca Ime-Usp
40:
18:Body of revolution
5007:Integral calculus
4959:978-0-07-150861-2
4950:Schaum's Outlines
4788:
4776:
4668:
4656:
4479:
4326:
4086:
4082:
4070:
4032:
3966:
3957:
3953:
3941:
3903:
3825:
3816:
3812:
3800:
3762:
3702:
3602:
3593:
3579:
3553:
3512:
3413:
3404:
3395:
3370:
3197:
3174:
3136:
3085:
3059:
3018:
2972:
2947:
2843:
2797:
2759:
2721:
2693:
2644:
2635:
2626:
2601:
2533:
2524:
2515:
2490:
2148:
2136:
2098:
2014:
2002:
1964:
1849:
1768:
1007:Shell integration
999:Shell integration
853:
542:and inner radius
93:), which may not
52:
16:(Redirected from
5019:
4993:
4992:
4963:
4933:
4908:
4907:
4887:
4881:
4880:
4860:
4829:Guldinus theorem
4814:
4812:
4811:
4806:
4804:
4789:
4787:
4786:
4781:
4777:
4775:
4767:
4759:
4749:
4748:
4739:
4736:
4715:
4710:
4694:
4693:
4669:
4667:
4666:
4661:
4657:
4655:
4647:
4639:
4629:
4628:
4619:
4616:
4595:
4590:
4574:
4573:
4554:-axis are given
4553:
4549:
4543:
4541:
4540:
4535:
4533:
4519:
4515:
4514:
4506:
4505:
4496:
4495:
4480:
4478:
4470:
4462:
4459:
4451:
4450:
4441:
4430:
4429:
4411:
4406:
4390:
4389:
4366:
4362:
4361:
4353:
4352:
4343:
4342:
4327:
4325:
4317:
4309:
4306:
4295:
4287:
4286:
4277:
4276:
4258:
4253:
4237:
4236:
4216:
4214:
4213:
4208:
4181:
4179:
4178:
4173:
4149:
4147:
4146:
4141:
4106:
4104:
4103:
4098:
4096:
4084:
4083:
4081:
4080:
4075:
4071:
4069:
4061:
4053:
4043:
4042:
4037:
4033:
4031:
4023:
4015:
4008:
3996:
3991:
3976:
3964:
3955:
3954:
3952:
3951:
3946:
3942:
3940:
3932:
3924:
3914:
3913:
3908:
3904:
3902:
3894:
3886:
3879:
3873:
3865:
3855:
3850:
3835:
3823:
3814:
3813:
3811:
3810:
3805:
3801:
3799:
3791:
3783:
3773:
3772:
3767:
3763:
3761:
3753:
3745:
3726:
3725:
3713:
3712:
3707:
3703:
3701:
3693:
3685:
3666:
3665:
3656:
3650:
3642:
3632:
3627:
3612:
3600:
3591:
3590:
3586:
3585:
3581:
3580:
3578:
3570:
3562:
3554:
3552:
3544:
3536:
3513:
3511:
3503:
3495:
3461:
3453:
3443:
3438:
3423:
3411:
3402:
3401:
3397:
3396:
3394:
3386:
3385:
3376:
3371:
3369:
3361:
3360:
3351:
3343:
3335:
3325:
3320:
3304:
3303:
3283:
3281:
3280:
3275:
3255:
3254:
3230:
3229:
3213:
3211:
3210:
3205:
3203:
3199:
3198:
3196:
3188:
3180:
3175:
3173:
3165:
3157:
3137:
3135:
3127:
3119:
3091:
3087:
3086:
3084:
3076:
3068:
3060:
3058:
3050:
3042:
3019:
3017:
3009:
3001:
2973:
2971:
2963:
2962:
2953:
2948:
2946:
2938:
2937:
2928:
2915:
2913:
2912:
2907:
2905:
2901:
2844:
2842:
2834:
2833:
2824:
2816:
2814:
2813:
2808:
2803:
2799:
2798:
2796:
2788:
2780:
2760:
2758:
2750:
2742:
2722:
2720:
2712:
2704:
2694:
2692:
2684:
2683:
2674:
2663:
2661:
2660:
2655:
2642:
2633:
2632:
2628:
2627:
2625:
2617:
2616:
2607:
2602:
2600:
2592:
2591:
2582:
2574:
2566:
2556:
2551:
2531:
2522:
2521:
2517:
2516:
2514:
2506:
2505:
2496:
2491:
2489:
2481:
2480:
2471:
2464:
2463:
2412:
2411:
2399:
2398:
2382:surface integral
2379:
2377:
2376:
2371:
2344:
2342:
2341:
2336:
2238:
2226:
2224:
2223:
2218:
2174:
2172:
2171:
2166:
2164:
2149:
2147:
2146:
2141:
2137:
2135:
2127:
2119:
2109:
2108:
2103:
2099:
2097:
2089:
2081:
2074:
2061:
2056:
2040:
2039:
2015:
2013:
2012:
2007:
2003:
2001:
1993:
1985:
1975:
1974:
1969:
1965:
1963:
1955:
1947:
1940:
1927:
1922:
1906:
1905:
1885:
1881:
1875:
1873:
1872:
1867:
1865:
1850:
1848:
1840:
1832:
1829:
1828:
1815:
1810:
1794:
1793:
1769:
1767:
1759:
1751:
1748:
1747:
1734:
1729:
1713:
1712:
1692:
1688:
1684:
1631:
1619:
1603:
1601:
1600:
1595:
1546:
1541:
1505:
1491:
1472:
1467:
1424:
1416:
1406:
1392:
1373:
1368:
1350:
1349:
1324:
1305:
1301:
1297:
1293:
1286:
1282:
1263:
1257:
1255:
1254:
1249:
1236:
1219:
1210:
1205:
1177:
1173:
1162:
1160:
1159:
1154:
1141:
1109:
1100:
1095:
1067:
1063:
1053:
1043:
1032:
1017:perpendicular to
989:
987:
986:
981:
969:
968:
947:
946:
924:
919:
896:
882:
864:
863:
854:
846:
843:
838:
802:
788:
769:
764:
721:
713:
703:
689:
670:
665:
647:
646:
623:Fubini's theorem
617:
613:
609:
594:
583:
579:
568:
564:
552:
541:
530:
519:
515:
504:
493:
487:
485:
484:
479:
466:
465:
446:
441:
416:
412:
401:
399:
398:
393:
380:
376:
375:
374:
353:
352:
328:
323:
298:
294:
284:
274:
263:
244:perpendicular to
238:Disc integration
211:
207:
203:
183:
166:
162:
118:is equal to the
53:
21:
5027:
5026:
5022:
5021:
5020:
5018:
5017:
5016:
4997:
4996:
4960:
4926:CliffsNotes.com
4920:
4917:
4912:
4911:
4904:
4888:
4884:
4877:
4861:
4857:
4852:
4820:
4802:
4801:
4782:
4768:
4760:
4758:
4754:
4753:
4744:
4740:
4738:
4732:
4711:
4706:
4695:
4689:
4685:
4682:
4681:
4662:
4648:
4640:
4638:
4634:
4633:
4624:
4620:
4618:
4612:
4591:
4586:
4575:
4569:
4565:
4561:
4559:
4556:
4555:
4551:
4547:
4531:
4530:
4510:
4501:
4497:
4491:
4487:
4471:
4463:
4461:
4455:
4446:
4442:
4437:
4425:
4421:
4417:
4413:
4407:
4402:
4391:
4385:
4381:
4378:
4377:
4357:
4348:
4344:
4338:
4334:
4318:
4310:
4308:
4302:
4291:
4282:
4278:
4272:
4268:
4264:
4260:
4254:
4249:
4238:
4232:
4228:
4224:
4222:
4219:
4218:
4187:
4184:
4183:
4155:
4152:
4151:
4120:
4117:
4116:
4113:
4094:
4093:
4076:
4062:
4054:
4052:
4048:
4047:
4038:
4024:
4016:
4014:
4010:
4009:
4007:
3992:
3987:
3974:
3973:
3947:
3933:
3925:
3923:
3919:
3918:
3909:
3895:
3887:
3885:
3881:
3880:
3878:
3866:
3861:
3851:
3846:
3833:
3832:
3806:
3792:
3784:
3782:
3778:
3777:
3768:
3754:
3746:
3744:
3740:
3739:
3721:
3717:
3708:
3694:
3686:
3684:
3680:
3679:
3661:
3657:
3655:
3643:
3638:
3628:
3623:
3610:
3609:
3571:
3563:
3561:
3545:
3537:
3535:
3504:
3496:
3494:
3475:
3471:
3467:
3463:
3454:
3449:
3439:
3434:
3421:
3420:
3387:
3381:
3377:
3375:
3362:
3356:
3352:
3350:
3349:
3345:
3336:
3331:
3321:
3316:
3305:
3299:
3295:
3291:
3289:
3286:
3285:
3250:
3246:
3225:
3221:
3219:
3216:
3215:
3189:
3181:
3179:
3166:
3158:
3156:
3128:
3120:
3118:
3102:
3098:
3077:
3069:
3067:
3051:
3043:
3041:
3010:
3002:
3000:
2981:
2977:
2964:
2958:
2954:
2952:
2939:
2933:
2929:
2927:
2925:
2922:
2921:
2852:
2848:
2835:
2829:
2825:
2823:
2821:
2818:
2817:
2789:
2781:
2779:
2751:
2743:
2741:
2713:
2705:
2703:
2702:
2698:
2685:
2679:
2675:
2673:
2671:
2668:
2667:
2618:
2612:
2608:
2606:
2593:
2587:
2583:
2581:
2580:
2576:
2567:
2562:
2552:
2547:
2507:
2501:
2497:
2495:
2482:
2476:
2472:
2470:
2469:
2465:
2426:
2422:
2407:
2403:
2394:
2390:
2388:
2385:
2384:
2350:
2347:
2346:
2234:
2232:
2229:
2228:
2182:
2179:
2178:
2162:
2161:
2142:
2128:
2120:
2118:
2114:
2113:
2104:
2090:
2082:
2080:
2076:
2075:
2073:
2057:
2052:
2041:
2035:
2031:
2028:
2027:
2008:
1994:
1986:
1984:
1980:
1979:
1970:
1956:
1948:
1946:
1942:
1941:
1939:
1923:
1918:
1907:
1901:
1897:
1893:
1891:
1888:
1887:
1883:
1879:
1863:
1862:
1841:
1833:
1831:
1824:
1820:
1811:
1806:
1795:
1789:
1785:
1782:
1781:
1760:
1752:
1750:
1743:
1739:
1730:
1725:
1714:
1708:
1704:
1700:
1698:
1695:
1694:
1690:
1686:
1666:
1644:
1642:Parametric form
1639:
1638:
1637:
1636:
1635:
1632:
1624:
1623:
1620:
1611:
1610:
1542:
1537:
1492:
1478:
1468:
1463:
1417:
1412:
1393:
1379:
1369:
1364:
1345:
1341:
1333:
1330:
1329:
1307:
1303:
1299:
1295:
1288:
1284:
1265:
1261:
1232:
1215:
1206:
1201:
1183:
1180:
1179:
1175:
1164:
1137:
1105:
1096:
1091:
1073:
1070:
1069:
1065:
1055:
1045:
1034:
1023:
1001:
995:
964:
960:
942:
938:
920:
915:
883:
869:
859:
855:
845:
839:
834:
789:
775:
765:
760:
714:
709:
690:
676:
666:
661:
642:
638:
630:
627:
626:
615:
611:
596:
585:
581:
570:
566:
554:
543:
532:
521:
517:
506:
495:
491:
461:
457:
442:
437:
422:
419:
418:
414:
403:
370:
366:
348:
344:
334:
330:
324:
319:
304:
301:
300:
296:
286:
276:
265:
254:
240:
226:
214:triple integral
209:
205:
201:
190:
175:
164:
160:
42:
28:
23:
22:
15:
12:
11:
5:
5025:
5015:
5014:
5009:
4995:
4994:
4975:
4958:
4934:
4932:on 2012-03-19.
4916:
4913:
4910:
4909:
4902:
4882:
4875:
4854:
4853:
4851:
4848:
4847:
4846:
4841:
4836:
4831:
4826:
4824:Gabriel's Horn
4819:
4816:
4800:
4796:
4793:
4785:
4780:
4774:
4771:
4766:
4763:
4757:
4752:
4747:
4743:
4735:
4731:
4728:
4725:
4722:
4719:
4714:
4709:
4705:
4701:
4698:
4696:
4692:
4688:
4684:
4683:
4680:
4676:
4673:
4665:
4660:
4654:
4651:
4646:
4643:
4637:
4632:
4627:
4623:
4615:
4611:
4608:
4605:
4602:
4599:
4594:
4589:
4585:
4581:
4578:
4576:
4572:
4568:
4564:
4563:
4529:
4525:
4522:
4518:
4513:
4509:
4504:
4500:
4494:
4490:
4486:
4483:
4477:
4474:
4469:
4466:
4458:
4454:
4449:
4445:
4440:
4436:
4433:
4428:
4424:
4420:
4416:
4410:
4405:
4401:
4397:
4394:
4392:
4388:
4384:
4380:
4379:
4376:
4372:
4369:
4365:
4360:
4356:
4351:
4347:
4341:
4337:
4333:
4330:
4324:
4321:
4316:
4313:
4305:
4301:
4298:
4294:
4290:
4285:
4281:
4275:
4271:
4267:
4263:
4257:
4252:
4248:
4244:
4241:
4239:
4235:
4231:
4227:
4226:
4206:
4203:
4200:
4197:
4194:
4191:
4171:
4168:
4165:
4162:
4159:
4139:
4136:
4133:
4130:
4127:
4124:
4112:
4109:
4092:
4089:
4079:
4074:
4068:
4065:
4060:
4057:
4051:
4046:
4041:
4036:
4030:
4027:
4022:
4019:
4013:
4006:
4003:
4000:
3995:
3990:
3986:
3982:
3979:
3977:
3975:
3972:
3969:
3963:
3960:
3950:
3945:
3939:
3936:
3931:
3928:
3922:
3917:
3912:
3907:
3901:
3898:
3893:
3890:
3884:
3877:
3872:
3869:
3864:
3860:
3854:
3849:
3845:
3841:
3838:
3836:
3834:
3831:
3828:
3822:
3819:
3809:
3804:
3798:
3795:
3790:
3787:
3781:
3776:
3771:
3766:
3760:
3757:
3752:
3749:
3743:
3738:
3735:
3732:
3729:
3724:
3720:
3716:
3711:
3706:
3700:
3697:
3692:
3689:
3683:
3678:
3675:
3672:
3669:
3664:
3660:
3654:
3649:
3646:
3641:
3637:
3631:
3626:
3622:
3618:
3615:
3613:
3611:
3608:
3605:
3599:
3596:
3589:
3584:
3577:
3574:
3569:
3566:
3560:
3557:
3551:
3548:
3543:
3540:
3534:
3531:
3528:
3525:
3522:
3519:
3516:
3510:
3507:
3502:
3499:
3493:
3490:
3487:
3484:
3481:
3478:
3474:
3470:
3466:
3460:
3457:
3452:
3448:
3442:
3437:
3433:
3429:
3426:
3424:
3422:
3419:
3416:
3410:
3407:
3400:
3393:
3390:
3384:
3380:
3374:
3368:
3365:
3359:
3355:
3348:
3342:
3339:
3334:
3330:
3324:
3319:
3315:
3311:
3308:
3306:
3302:
3298:
3294:
3293:
3273:
3270:
3267:
3264:
3261:
3258:
3253:
3249:
3245:
3242:
3239:
3236:
3233:
3228:
3224:
3202:
3195:
3192:
3187:
3184:
3178:
3172:
3169:
3164:
3161:
3155:
3152:
3149:
3146:
3143:
3140:
3134:
3131:
3126:
3123:
3117:
3114:
3111:
3108:
3105:
3101:
3097:
3094:
3090:
3083:
3080:
3075:
3072:
3066:
3063:
3057:
3054:
3049:
3046:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3016:
3013:
3008:
3005:
2999:
2996:
2993:
2990:
2987:
2984:
2980:
2976:
2970:
2967:
2961:
2957:
2951:
2945:
2942:
2936:
2932:
2904:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2851:
2847:
2841:
2838:
2832:
2828:
2806:
2802:
2795:
2792:
2787:
2784:
2778:
2775:
2772:
2769:
2766:
2763:
2757:
2754:
2749:
2746:
2740:
2737:
2734:
2731:
2728:
2725:
2719:
2716:
2711:
2708:
2701:
2697:
2691:
2688:
2682:
2678:
2653:
2650:
2647:
2641:
2638:
2631:
2624:
2621:
2615:
2611:
2605:
2599:
2596:
2590:
2586:
2579:
2573:
2570:
2565:
2561:
2555:
2550:
2546:
2542:
2539:
2536:
2530:
2527:
2520:
2513:
2510:
2504:
2500:
2494:
2488:
2485:
2479:
2475:
2468:
2462:
2459:
2456:
2453:
2450:
2447:
2444:
2441:
2438:
2435:
2432:
2429:
2425:
2421:
2418:
2415:
2410:
2406:
2402:
2397:
2393:
2369:
2366:
2363:
2360:
2357:
2354:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2237:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2160:
2156:
2153:
2145:
2140:
2134:
2131:
2126:
2123:
2117:
2112:
2107:
2102:
2096:
2093:
2088:
2085:
2079:
2071:
2068:
2065:
2060:
2055:
2051:
2047:
2044:
2042:
2038:
2034:
2030:
2029:
2026:
2022:
2019:
2011:
2006:
2000:
1997:
1992:
1989:
1983:
1978:
1973:
1968:
1962:
1959:
1954:
1951:
1945:
1937:
1934:
1931:
1926:
1921:
1917:
1913:
1910:
1908:
1904:
1900:
1896:
1895:
1861:
1857:
1854:
1847:
1844:
1839:
1836:
1827:
1823:
1819:
1814:
1809:
1805:
1801:
1798:
1796:
1792:
1788:
1784:
1783:
1780:
1776:
1773:
1766:
1763:
1758:
1755:
1746:
1742:
1738:
1733:
1728:
1724:
1720:
1717:
1715:
1711:
1707:
1703:
1702:
1657:. 15th century
1643:
1640:
1633:
1626:
1625:
1621:
1614:
1613:
1612:
1608:
1607:
1606:
1605:
1593:
1590:
1587:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1545:
1540:
1536:
1532:
1529:
1526:
1523:
1520:
1516:
1513:
1509:
1504:
1501:
1498:
1495:
1490:
1487:
1484:
1481:
1477:
1471:
1466:
1462:
1458:
1455:
1452:
1449:
1446:
1442:
1439:
1435:
1432:
1428:
1423:
1420:
1415:
1411:
1405:
1402:
1399:
1396:
1391:
1388:
1385:
1382:
1378:
1372:
1367:
1363:
1359:
1356:
1353:
1348:
1344:
1340:
1337:
1247:
1243:
1240:
1235:
1231:
1228:
1225:
1222:
1218:
1214:
1209:
1204:
1200:
1196:
1193:
1190:
1187:
1152:
1148:
1145:
1140:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1108:
1104:
1099:
1094:
1090:
1086:
1083:
1080:
1077:
1044:and the lines
997:Main article:
994:
991:
979:
976:
972:
967:
963:
959:
956:
953:
950:
945:
941:
937:
934:
931:
928:
923:
918:
914:
910:
907:
904:
901:
895:
892:
889:
886:
881:
878:
875:
872:
868:
862:
858:
852:
849:
842:
837:
833:
829:
826:
823:
820:
817:
813:
810:
806:
801:
798:
795:
792:
787:
784:
781:
778:
774:
768:
763:
759:
755:
752:
749:
746:
743:
739:
736:
732:
729:
725:
720:
717:
712:
708:
702:
699:
696:
693:
688:
685:
682:
679:
675:
669:
664:
660:
656:
653:
650:
645:
641:
637:
634:
477:
473:
470:
464:
460:
456:
453:
450:
445:
440:
436:
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275:and the lines
236:Main article:
225:
222:
189:
186:
150:volume element
26:
9:
6:
4:
3:
2:
5024:
5013:
5010:
5008:
5005:
5004:
5002:
4990:
4989:
4984:
4981:
4976:
4973:
4970:, p. 244, at
4969:
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4947:
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4939:
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4927:
4923:
4919:
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4903:0-07-014615-2
4899:
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4886:
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4876:81-7141-967-4
4872:
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4566:
4550:-axis or the
4544:
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2918:cross product
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2110:
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2100:
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2069:
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2063:
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2049:
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2043:
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2032:
2024:
2020:
2017:
2009:
2004:
1998:
1995:
1990:
1987:
1981:
1976:
1971:
1966:
1960:
1957:
1952:
1949:
1943:
1935:
1932:
1929:
1924:
1919:
1915:
1911:
1909:
1902:
1898:
1882:-axis or the
1876:
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1716:
1709:
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1689:-axis or the
1682:
1678:
1674:
1670:
1664:
1656:
1655:Paolo Uccello
1652:
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790:
782:
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766:
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710:
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691:
683:
677:
673:
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658:
654:
651:
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639:
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632:
624:
619:
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604:
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592:
588:
577:
573:
562:
558:
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546:
539:
535:
528:
524:
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502:
498:
488:
475:
471:
468:
462:
454:
448:
443:
438:
434:
430:
427:
424:
410:
406:
389:
385:
382:
377:
371:
363:
357:
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341:
335:
331:
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320:
316:
312:
309:
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293:
289:
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199:
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185:
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159:
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148:
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137:
133:
129:
125:
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117:
112:
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104:
100:
96:
92:
91:
86:
85:straight line
82:
78:
74:
70:
66:
58:
37:
32:
19:
4986:
4972:Google Books
4965:
4945:
4938:Ayres, Frank
4930:the original
4925:
4892:
4885:
4865:
4858:
4834:Pseudosphere
4545:
4114:
2665:
2176:
1877:
1680:
1676:
1672:
1668:
1660:
1327:
1320:
1316:
1312:
1308:
1290:
1278:
1274:
1270:
1266:
1264:with height
1259:
1169:
1165:
1060:
1056:
1050:
1046:
1039:
1035:
1028:
1024:
1021:
1016:
1012:
1010:
620:
606:
602:
598:
590:
586:
575:
571:
560:
556:
548:
544:
537:
533:
526:
522:
511:
507:
500:
496:
489:
408:
404:
291:
287:
281:
277:
270:
266:
259:
255:
252:
247:
243:
241:
191:
180:
177:
154:line segment
142:
140:
113:
106:
88:
83:around some
81:plane figure
75:obtained by
73:solid figure
68:
62:
4967:online copy
1013:parallel to
505:on top and
248:parallel to
224:Disc method
194:disc method
169:cylindrical
147:dimensional
145:is a three-
5001:Categories
4915:References
4111:Polar form
1663:parametric
1064:about the
295:about the
99:generatrix
4988:MathWorld
4795:θ
4773:θ
4734:θ
4730:
4721:π
4713:β
4708:α
4704:∫
4675:θ
4653:θ
4614:θ
4610:
4601:π
4593:β
4588:α
4584:∫
4524:θ
4512:θ
4508:
4485:π
4476:θ
4457:θ
4453:
4439:θ
4435:
4419:π
4409:β
4404:α
4400:∫
4371:θ
4359:θ
4355:
4332:π
4329:−
4323:θ
4304:θ
4300:
4293:θ
4289:
4266:π
4256:β
4251:α
4247:∫
4202:≥
4196:θ
4170:β
4167:≤
4164:θ
4161:≤
4158:α
4135:θ
4002:π
3985:∫
3962:θ
3871:π
3859:∫
3844:∫
3821:θ
3734:θ
3728:
3674:θ
3668:
3648:π
3636:∫
3621:∫
3598:θ
3530:θ
3524:
3489:θ
3483:
3459:π
3447:∫
3432:∫
3409:θ
3392:θ
3389:∂
3379:∂
3373:×
3364:∂
3354:∂
3341:π
3329:∫
3314:∫
3263:θ
3257:
3238:θ
3232:
3151:θ
3145:
3113:θ
3107:
3036:θ
3030:
2995:θ
2989:
2969:θ
2966:∂
2956:∂
2950:×
2941:∂
2931:∂
2890:θ
2884:
2869:θ
2863:
2854:−
2840:θ
2837:∂
2827:∂
2771:θ
2765:
2733:θ
2727:
2687:∂
2677:∂
2640:θ
2623:θ
2620:∂
2610:∂
2604:×
2595:∂
2585:∂
2572:π
2560:∫
2545:∫
2529:θ
2512:θ
2509:∂
2499:∂
2493:×
2484:∂
2474:∂
2458:π
2443:×
2424:∬
2405:∬
2368:π
2362:≤
2359:θ
2356:≤
2333:⟩
2312:θ
2306:
2282:θ
2276:
2258:⟨
2249:θ
2215:⟩
2185:⟨
2067:π
2050:∫
1933:π
1916:∫
1818:π
1804:∫
1737:π
1723:∫
1567:−
1535:∫
1531:π
1476:∫
1461:∫
1457:π
1434:θ
1422:π
1410:∫
1377:∫
1362:∫
1343:∭
1199:∫
1195:π
1123:−
1089:∫
1085:π
949:−
913:∫
909:π
867:‖
832:∫
828:π
773:∫
758:∫
754:π
731:θ
719:π
707:∫
674:∫
659:∫
640:∭
435:∫
431:π
355:−
317:∫
313:π
95:intersect
4946:Calculus
4944:(2008).
4818:See also
3588:‖
3583:⟩
3473:⟨
3465:‖
3399:‖
3347:‖
3201:⟩
3100:⟨
3089:⟩
2979:⟨
2920:yields
2903:⟩
2850:⟨
2801:⟩
2700:⟨
2630:‖
2578:‖
2519:‖
2467:‖
1294:, where
565:, where
494:between
196:and the
128:centroid
77:rotating
65:geometry
1302:), and
580:), and
122:of the
103:surface
5012:Solids
4956:
4900:
4873:
4844:Ungula
4150:where
4085:
3965:
3956:
3824:
3815:
3601:
3592:
3412:
3403:
2643:
2634:
2532:
2523:
172:volume
158:length
124:circle
120:length
116:volume
4850:Notes
2345:with
1665:form
1172:) = 0
529:) = 0
411:) = 0
87:(the
71:is a
4954:ISBN
4898:ISBN
4871:ISBN
4182:and
1315:) −
1273:) −
1054:and
1033:and
614:and
285:and
264:and
156:(of
132:area
97:the
67:, a
4727:cos
4607:sin
4499:cos
4444:cos
4432:sin
4346:sin
4297:cos
4280:sin
3719:sin
3659:cos
3521:sin
3480:cos
3248:cos
3223:sin
3142:sin
3104:cos
3027:sin
2986:cos
2881:cos
2860:sin
2762:sin
2724:cos
2303:sin
2273:cos
1163:If
402:If
216:in
174:of
138:).
63:In
5003::
4985:.
4948:.
4940:;
4924:.
1683:))
1675:),
1291:rh
1289:2π
1059:=
1049:=
607:dy
605:)
559:−
555:π(
290:=
280:=
210:δx
206:δx
202:δx
141:A
111:.
79:a
4991:.
4974:)
4964:(
4962:.
4906:.
4879:.
4799:,
4792:d
4784:2
4779:)
4770:d
4765:r
4762:d
4756:(
4751:+
4746:2
4742:r
4724:r
4718:2
4700:=
4691:y
4687:A
4679:,
4672:d
4664:2
4659:)
4650:d
4645:r
4642:d
4636:(
4631:+
4626:2
4622:r
4604:r
4598:2
4580:=
4571:x
4567:A
4552:y
4548:x
4528:.
4521:d
4517:)
4503:3
4493:3
4489:r
4482:+
4473:d
4468:r
4465:d
4448:2
4427:2
4423:r
4415:(
4396:=
4387:y
4383:V
4375:,
4368:d
4364:)
4350:3
4340:3
4336:r
4320:d
4315:r
4312:d
4284:2
4274:2
4270:r
4262:(
4243:=
4234:x
4230:V
4205:0
4199:)
4193:(
4190:f
4138:)
4132:(
4129:f
4126:=
4123:r
4091:t
4088:d
4078:2
4073:)
4067:t
4064:d
4059:y
4056:d
4050:(
4045:+
4040:2
4035:)
4029:t
4026:d
4021:x
4018:d
4012:(
4005:y
3999:2
3994:b
3989:a
3981:=
3971:t
3968:d
3959:d
3949:2
3944:)
3938:t
3935:d
3930:y
3927:d
3921:(
3916:+
3911:2
3906:)
3900:t
3897:d
3892:x
3889:d
3883:(
3876:y
3868:2
3863:0
3853:b
3848:a
3840:=
3830:t
3827:d
3818:d
3808:2
3803:)
3797:t
3794:d
3789:y
3786:d
3780:(
3775:+
3770:2
3765:)
3759:t
3756:d
3751:x
3748:d
3742:(
3737:)
3731:(
3723:2
3715:+
3710:2
3705:)
3699:t
3696:d
3691:x
3688:d
3682:(
3677:)
3671:(
3663:2
3653:y
3645:2
3640:0
3630:b
3625:a
3617:=
3607:t
3604:d
3595:d
3576:t
3573:d
3568:y
3565:d
3559:y
3556:,
3550:t
3547:d
3542:x
3539:d
3533:)
3527:(
3518:y
3515:,
3509:t
3506:d
3501:x
3498:d
3492:)
3486:(
3477:y
3469:y
3456:2
3451:0
3441:b
3436:a
3428:=
3418:t
3415:d
3406:d
3383:r
3367:t
3358:r
3338:2
3333:0
3323:b
3318:a
3310:=
3301:x
3297:A
3272:1
3269:=
3266:)
3260:(
3252:2
3244:+
3241:)
3235:(
3227:2
3194:t
3191:d
3186:y
3183:d
3177:,
3171:t
3168:d
3163:x
3160:d
3154:)
3148:(
3139:,
3133:t
3130:d
3125:x
3122:d
3116:)
3110:(
3096:y
3093:=
3082:t
3079:d
3074:y
3071:d
3065:y
3062:,
3056:t
3053:d
3048:x
3045:d
3039:)
3033:(
3024:y
3021:,
3015:t
3012:d
3007:x
3004:d
2998:)
2992:(
2983:y
2975:=
2960:r
2944:t
2935:r
2899:0
2896:,
2893:)
2887:(
2878:y
2875:,
2872:)
2866:(
2857:y
2846:=
2831:r
2805:,
2794:t
2791:d
2786:x
2783:d
2777:,
2774:)
2768:(
2756:t
2753:d
2748:y
2745:d
2739:,
2736:)
2730:(
2718:t
2715:d
2710:y
2707:d
2696:=
2690:t
2681:r
2652:.
2649:t
2646:d
2637:d
2614:r
2598:t
2589:r
2569:2
2564:0
2554:b
2549:a
2541:=
2538:t
2535:d
2526:d
2503:r
2487:t
2478:r
2461:]
2455:2
2452:,
2449:0
2446:[
2440:]
2437:b
2434:,
2431:a
2428:[
2420:=
2417:S
2414:d
2409:S
2401:=
2396:x
2392:A
2365:2
2353:0
2330:)
2327:t
2324:(
2321:x
2318:,
2315:)
2309:(
2300:)
2297:t
2294:(
2291:y
2288:,
2285:)
2279:(
2270:)
2267:t
2264:(
2261:y
2255:=
2252:)
2246:,
2243:t
2240:(
2236:r
2212:)
2209:t
2206:(
2203:y
2200:,
2197:)
2194:t
2191:(
2188:x
2159:.
2155:t
2152:d
2144:2
2139:)
2133:t
2130:d
2125:y
2122:d
2116:(
2111:+
2106:2
2101:)
2095:t
2092:d
2087:x
2084:d
2078:(
2070:x
2064:2
2059:b
2054:a
2046:=
2037:y
2033:A
2025:,
2021:t
2018:d
2010:2
2005:)
1999:t
1996:d
1991:y
1988:d
1982:(
1977:+
1972:2
1967:)
1961:t
1958:d
1953:x
1950:d
1944:(
1936:y
1930:2
1925:b
1920:a
1912:=
1903:x
1899:A
1884:y
1880:x
1860:.
1856:t
1853:d
1846:t
1843:d
1838:y
1835:d
1826:2
1822:x
1813:b
1808:a
1800:=
1791:y
1787:V
1779:,
1775:t
1772:d
1765:t
1762:d
1757:x
1754:d
1745:2
1741:y
1732:b
1727:a
1719:=
1710:x
1706:V
1691:y
1687:x
1681:t
1679:(
1677:y
1673:t
1671:(
1669:x
1667:(
1592:.
1589:r
1586:d
1582:)
1579:)
1576:r
1573:(
1570:g
1564:)
1561:r
1558:(
1555:f
1552:(
1549:r
1544:b
1539:a
1528:2
1525:=
1522:r
1519:d
1515:z
1512:d
1508:r
1503:)
1500:r
1497:(
1494:f
1489:)
1486:r
1483:(
1480:g
1470:b
1465:a
1454:2
1451:=
1448:r
1445:d
1441:z
1438:d
1431:d
1427:r
1419:2
1414:0
1404:)
1401:r
1398:(
1395:f
1390:)
1387:r
1384:(
1381:g
1371:b
1366:a
1358:=
1355:V
1352:d
1347:D
1339:=
1336:V
1323:)
1321:x
1319:(
1317:g
1313:x
1311:(
1309:f
1304:h
1300:x
1296:r
1285:y
1281:)
1279:x
1277:(
1275:g
1271:x
1269:(
1267:f
1262:x
1246:.
1242:x
1239:d
1234:|
1230:)
1227:x
1224:(
1221:f
1217:|
1213:x
1208:b
1203:a
1192:2
1189:=
1186:V
1176:y
1170:x
1168:(
1166:g
1151:.
1147:x
1144:d
1139:|
1135:)
1132:x
1129:(
1126:g
1120:)
1117:x
1114:(
1111:f
1107:|
1103:x
1098:b
1093:a
1082:2
1079:=
1076:V
1066:y
1061:b
1057:x
1051:a
1047:x
1042:)
1040:x
1038:(
1036:g
1031:)
1029:x
1027:(
1025:f
978:z
975:d
971:)
966:2
962:)
958:z
955:(
952:g
944:2
940:)
936:z
933:(
930:f
927:(
922:b
917:a
906:=
903:z
900:d
894:)
891:z
888:(
885:f
880:)
877:z
874:(
871:g
861:2
857:r
851:2
848:1
841:b
836:a
825:2
822:=
819:z
816:d
812:r
809:d
805:r
800:)
797:z
794:(
791:f
786:)
783:z
780:(
777:g
767:b
762:a
751:2
748:=
745:z
742:d
738:r
735:d
728:d
724:r
716:2
711:0
701:)
698:z
695:(
692:f
687:)
684:z
681:(
678:g
668:b
663:a
655:=
652:V
649:d
644:D
636:=
633:V
616:b
612:a
603:y
601:(
599:f
597:π
593:)
591:y
589:(
587:g
582:r
578:)
576:y
574:(
572:f
567:R
563:)
561:r
557:R
551:)
549:y
547:(
545:g
540:)
538:y
536:(
534:f
527:y
525:(
523:g
518:y
514:)
512:y
510:(
508:g
503:)
501:y
499:(
497:f
492:y
476:.
472:y
469:d
463:2
459:)
455:y
452:(
449:f
444:b
439:a
428:=
425:V
415:y
409:y
407:(
405:g
390:.
386:y
383:d
378:|
372:2
368:)
364:y
361:(
358:g
350:2
346:)
342:y
339:(
336:f
332:|
326:b
321:a
310:=
307:V
297:y
292:b
288:y
282:a
278:y
273:)
271:y
269:(
267:g
262:)
260:y
258:(
256:f
181:w
178:r
176:π
165:r
161:w
134:(
59:)
20:)
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