Example 2.3.17: Volume of a sphere

The ball is the region $0\le r\le b$, $0\le \theta \le \pi$, $0\le \varphi <2\pi$, with the volume element of Eq. (2.3.41). Again the limits are constant and the integrand factorises:

A useful check: the ball can be built from thin spherical shells of radius $r$, area $4\pi r^2$ (the result of Example 2.3.16) and thickness $dr$, like the layers of an onion, so $V={\int }_0^{b}4\pi r^{2}dr=\frac{4}{3}\pi b^3$. Conversely $dV/db=4\pi b^2$: the derivative of the volume with respect to the radius is the area.