A volume integral takes the form

 (1.1)

where V is some volume and dV = dx dy dz is a small volume element. The volume element is sometimes written d³r, or even dτ. As an example of a volume integral, let us evaluate the centre of gravity of a solid hemisphere of radius a. The height of the centre of gravity is given by

 (1.2)

The bottom integral is simply the volume of the hemisphere, which is 2πa³/3. The top integral is most easily evaluated in spherical polar coordinates, for which z = r cos θ and dV = r² sin θ dr dθ dϕ. Thus,

 (1.3)

giving

 (1.4)