An example

Find the volume of a sphere of radius a. The volume is simply the integral of the volume element over the entire region, so the volume is

$$\int_W dV = \int_0^{2\pi} \int_0^{\pi} \int_0^a \rho^2 \sin \phi d\rho d\phi d\theta$$

$$= \int_0^{2\pi} \int_0^{\pi} \left. \frac{1}{3}\rho^3 \sin \phi \right\vert _0^a d\phi d\theta$$

$$= \frac{a^3}{3} \int_0^{2\pi} \left. -\cos \phi \right\vert _0^{\pi} d\theta$$

$$= \frac{2a^3}{3} \left. \theta \right\vert _0^{2\pi} = \frac{4\pi}{3}a^3.$$