Question

Find the volume inside the sphere x^2 + y^2 + z^2 =16 and outside the cylinder x^2 + y^2 =4.

Answer 1

$0\le \theta\le 2\pi,2\le r\le 4$

Volume inside sphere but outside cylinder:

$V=2\iint\sqrt{16-x^2-y^2}dA=2\int^{2\pi}_0\int^4_2r(\sqrt{16-r^2})drd\theta=$

$=-\frac{2}{3}\int^{2\pi}_0(16-r^2)^{3/2}|^4_2d\theta=2\frac{(2\sqrt3)^{3}}{3}\int^{2\pi}_0d\theta=16\sqrt3\cdot2\pi=32\pi\sqrt3$