spherical coordinates

Distance $\rho$ from the origin plus two angles $\theta,\phi$ ; volume element $dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$ .

Spherical coordinates address a point by straight-line distance $\rho$ from the origin and two angles: $\theta$ , the same angle cylindrical coordinates use in the $xy$ -plane, and $\phi$ , the angle down from the positive $z$ -axis, restricted to $0\le\phi\le\pi$ . A sphere centered at the origin becomes the single equation $\rho=R$ . The volume element $dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$ carries two distortion factors: $\rho^2$ because a thin radial shell's surface area grows with the square of its radius, and $\sin\phi$ because circles of constant latitude shrink to a point at the poles.

Examples

The solid hemisphere $x^2+y^2+z^2\le4$ , $z\ge0$ has volume $\int_0^{2\pi}\!\int_0^{\pi/2}\!\int_0^2\rho^2\sin\phi\,d\rho\,d\phi\,d\theta=\tfrac{16\pi}{3}$ .