Volume of a solid between sphere and cylinder Find the volume of a solid inside a sphere $x^{2} + y^{2} + z^{2} = 4$ and the cylinder $x^{2} + y^{2} + 2y = 0$
Any help is appreciated

The simplest way is evaluate the integral in cylindrical polar coordinate.
$$(r,\theta,z)\quad\mapsto\quad (x,y,z) = (r\cos\theta,r\sin\theta,z)$$

 $$\begin{align}\verb/Volume/ &= \int_{-\pi}^0 \left[\int_0^{-2\sin\theta} 2\sqrt{4-r^2}\, r dr \right] d\theta = \int_{-\pi}^0  \left[ -\frac{2}{3}\sqrt{4-r^2}^3 \right]_0^{-2\sin\theta} d\theta\\ &= \frac{16}{3}\int_0^\pi (1 - |\cos\theta|^3)d\theta = \frac{16}{3}\left(\pi - \frac{4}{3}\right)\\ \end{align}$$

