H = ∫ − r r 4 3 π ( r 2 − x 2 ) 3 d x = ∫ − r r 4 3 π r 3 ( 1 − x 2 r 2 ) 3 d x = 4 3 π r 3 ∫ − r r ( 1 − x 2 r 2 ) 3 d x {\displaystyle {\begin{aligned}H&=\int _{-r}^{r}{\frac {4}{3}}\pi \left({\sqrt {r^{2}-x^{2}}}\right)^{3}dx&=\int _{-r}^{r}{\frac {4}{3}}\pi r^{3}\left({\sqrt {1-{\frac {x^{2}}{r^{2}}}}}\right)^{3}dx&={\frac {4}{3}}\pi r^{3}\int _{-r}^{r}\left({\sqrt {1-{\frac {x^{2}}{r^{2}}}}}\right)^{3}dx\end{aligned}}}
Let x = r sin ( u ) . {\displaystyle x=r\sin(u).} As a result, arcsin ( x r ) = u {\displaystyle \arcsin \left({\frac {x}{r}}\right)=u} and d x = r cos ( u ) d u . {\displaystyle dx=r\cos(u)du.}
H = 4 3 π r 3 ∫ a r c s i n ( − r r ) a r c s i n ( r r ) ( 1 − ( r sin ( u ) ) 2 r 2 ) 3 r cos ( u ) d u = 4 3 π r 3 ( r ) ∫ − π 2 π 2 ( 1 − sin 2 ( u ) ) 3 cos ( u ) d u = 4 3 π r 4 ∫ − π 2 π 2 ( c o s ( u ) ) 3 cos ( u ) d u = 4 3 π r 4 ∫ − π 2 π 2 c o s 4 ( u ) d u = 4 3 π r 4 ∫ − π 2 π 2 [ 1 8 ( 3 + 4 cos ( 2 u ) + cos ( 4 u ) ) ] d u = 4 3 ( 1 8 ) π r 4 ∫ − π 2 π 2 ( 3 + 4 cos ( 2 u ) + cos ( 4 u ) ) d u = 1 6 π r 4 [ 3 u + 2 sin ( 2 u ) + 1 4 sin ( 4 u ) ] u = − π 2 π 2 = 1 6 π r 4 [ ( 3 ( π 2 ) + 2 sin ( π ) + 1 4 sin ( 2 π ) ) − ( 3 ( − π 2 ) + 2 sin ( − π ) + 1 4 sin ( − 2 π ) ) ] = 1 6 π r 4 [ ( 3 ( π 2 ) ) − ( 3 ( − π 2 ) ) ] = 1 6 π r 4 [ ( 3 ( π 2 ) ) + ( 3 ( π 2 ) ) ] = 1 6 π r 4 ( 3 π 2 ) ( 2 ) = 1 6 π r 4 ( 3 π ) = 1 2 π 2 r 4 {\displaystyle {\begin{aligned}H&={\frac {4}{3}}\pi r^{3}\int _{arcsin\left({\frac {-r}{r}}\right)}^{arcsin\left({\frac {r}{r}}\right)}\left({\sqrt {1-{\frac {(r\sin(u))^{2}}{r^{2}}}}}\right)^{3}r\cos(u)du\\&={\frac {4}{3}}\pi r^{3}(r)\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}\left({\sqrt {1-\sin ^{2}(u)}}\right)^{3}\cos(u)du\\&={\frac {4}{3}}\pi r^{4}\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}(cos(u))^{3}\cos(u)du\\&={\frac {4}{3}}\pi r^{4}\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}cos^{4}(u)du\\&={\frac {4}{3}}\pi r^{4}\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}\left[{\frac {1}{8}}(3+4\cos(2u)+\cos(4u))\right]du\\&={\frac {4}{3}}\left({\frac {1}{8}}\right)\pi r^{4}\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}(3+4\cos(2u)+\cos(4u))du\\&={\frac {1}{6}}\pi r^{4}\left[3u+2\sin(2u)+{\frac {1}{4}}\sin(4u)\right]_{u=-{\frac {\pi }{2}}}^{\frac {\pi }{2}}\\&={\frac {1}{6}}\pi r^{4}\left[\left(3\left({\frac {\pi }{2}}\right)+2\sin(\pi )+{\frac {1}{4}}\sin(2\pi )\right)-\left(3\left(-{\frac {\pi }{2}}\right)+2\sin(-\pi )+{\frac {1}{4}}\sin(-2\pi )\right)\right]\\&={\frac {1}{6}}\pi r^{4}\left[\left(3\left({\frac {\pi }{2}}\right)\right)-\left(3\left(-{\frac {\pi }{2}}\right)\right)\right]\\&={\frac {1}{6}}\pi r^{4}\left[\left(3\left({\frac {\pi }{2}}\right)\right)+\left(3\left({\frac {\pi }{2}}\right)\right)\right]\\&={\frac {1}{6}}\pi r^{4}\left({\frac {3\pi }{2}}\right)(2)\\&={\frac {1}{6}}\pi r^{4}(3\pi )\\&={\frac {1}{2}}\pi ^{2}r^{4}\end{aligned}}}
![{\displaystyle {\begin{aligned}H&={\frac {4}{3}}\pi r^{3}\int _{arcsin\left({\frac {-r}{r}}\right)}^{arcsin\left({\frac {r}{r}}\right)}\left({\sqrt {1-{\frac {(r\sin(u))^{2}}{r^{2}}}}}\right)^{3}r\cos(u)du\\&={\frac {4}{3}}\pi r^{3}(r)\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}\left({\sqrt {1-\sin ^{2}(u)}}\right)^{3}\cos(u)du\\&={\frac {4}{3}}\pi r^{4}\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}(cos(u))^{3}\cos(u)du\\&={\frac {4}{3}}\pi r^{4}\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}cos^{4}(u)du\\&={\frac {4}{3}}\pi r^{4}\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}\left[{\frac {1}{8}}(3+4\cos(2u)+\cos(4u))\right]du\\&={\frac {4}{3}}\left({\frac {1}{8}}\right)\pi r^{4}\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}(3+4\cos(2u)+\cos(4u))du\\&={\frac {1}{6}}\pi r^{4}\left[3u+2\sin(2u)+{\frac {1}{4}}\sin(4u)\right]_{u=-{\frac {\pi }{2}}}^{\frac {\pi }{2}}\\&={\frac {1}{6}}\pi r^{4}\left[\left(3\left({\frac {\pi }{2}}\right)+2\sin(\pi )+{\frac {1}{4}}\sin(2\pi )\right)-\left(3\left(-{\frac {\pi }{2}}\right)+2\sin(-\pi )+{\frac {1}{4}}\sin(-2\pi )\right)\right]\\&={\frac {1}{6}}\pi r^{4}\left[\left(3\left({\frac {\pi }{2}}\right)\right)-\left(3\left(-{\frac {\pi }{2}}\right)\right)\right]\\&={\frac {1}{6}}\pi r^{4}\left[\left(3\left({\frac {\pi }{2}}\right)\right)+\left(3\left({\frac {\pi }{2}}\right)\right)\right]\\&={\frac {1}{6}}\pi r^{4}\left({\frac {3\pi }{2}}\right)(2)\\&={\frac {1}{6}}\pi r^{4}(3\pi )\\&={\frac {1}{2}}\pi ^{2}r^{4}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68dceaa4ca91c04314b664d0ce8356016bf278d2)