Volume integrals

A volume integral takes the form

$$\int\!\int\!\int_V f(x,y,z) dV,$$ (89)

where $V$ is some volume, and $dV = dx dy dz$ is a small volume element. The volume element is sometimes written $d^3{\bf r}$, or even $d\tau$. As an example of a volume integral, let us evaluate the centre of gravity of a solid hemisphere of radius $a$ (centered on the origin). The height of the centre of gravity is given by

$$\overline{z} = \left. \int\!\int\!\int z dV\right/ \int\!\int\!\int dV.$$ (90)

The bottom integral is simply the volume of the hemisphere, which is $2\pi a^3/3$. The top integral is most easily evaluated in spherical polar coordinates, for which $z= r \cos\theta$ and $dV = r^2 \sin\theta dr d\theta d\phi$. Thus,

$$\int\int \int z dV = \int_0^a dr\int_0^{\pi/2} d\theta \int_0^{2\pi} d\phi r \cos\theta r^2 \sin\theta$$

$$= \int_0^a r^3 dr \int_0^{\pi/2} \sin\theta \cos\theta d\theta \int_0^{2\pi} d\phi = \frac{\pi a^4}{4},$$ (91)

giving

$$\overline{z} = \frac{ \pi a^4}{4}\frac{3}{2\pi a^3}= \frac{3 a}{8}.$$ (92)