Answer to Question #114347 in Calculus for kaveri

Question

find out the volume of hemi sphere of radius R using spherical polar co ordinate system.

Answer 1

For hemisphere $\phi\in[0,\pi],\,\,\theta\in[0,\pi],\,\,\rho=r\in[0,R]$

The Jacobian for spherical coordinates is given by $J=r^2\sin\theta$

$V=\int\limits_0^\pi \int\limits_0^\pi \int\limits_0^R r^2\sin\theta\,dr\,d\theta\,d\phi=\frac{\pi R^3}{3}\int\limits_0^\pi \sin\theta\,d\theta=\frac{2}{3}\pi R^3$