Question

Use polar coordinates to find the volume of the given solid. inside the sphere x2 + y2 + z2 = 25 and outside the cylinder x2 + y2 = 1

Answer 1

$\begin{cases}x=\rho\cos\theta\sin\varphi\\y=\rho\sin\theta\sin\varphi\\z=\rho\cos\varphi\end{cases}$

Denote the space between the cylinder and sphere by $E$. Then

$\displaystyle\iiint_E\mathrm dV=\int_{\theta=0}^{\theta=2\pi}\int_{\varphi=\tan^{-1}\frac1{\sqrt{24}}}^{\varphi=\pi-\tan^{-1}\frac1{\sqrt{24}}}\int_{\rho=\csc\varphi}^{\rho=5}\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm d\theta$
$=64\pi\sqrt6$