The volume element in spherical coordinates

A blowup of a piece of a sphere is shown below.

Using a little trigonometry and geometry, we can measure the sides of this element (as shown in the figure) and compute the volume as

$$\Delta V \approx (\rho \sin \phi \Delta \theta) (\rho \Delta \phi) (\Delta \rho)$$

so that, in the infinitesimal limit,

$$dV = \rho^2 \sin \phi d\phi d\theta d\rho.$$