A volume integral accumulates a scalar quantity throughout a three-dimensional region $V$:

$$\iiint_V f(x,y,z),dV.$$

When $f=1$, the integral gives the geometric volume of $V$.

Iterated integration

A triple integral can often be evaluated one variable at a time. For a rectangular box,

$$\iiint_V f,dV

\int_a^b\int_c^d\int_e^g f(x,y,z),dz,dy,dx.$$

For more general regions, the inner bounds may depend on the outer variables. Each choice of integration order corresponds to slicing the volume in a different way.

Accumulating a density

If $\rho(x,y,z)$ is a mass density, then the total mass is

$$m=\iiint_V \rho(x,y,z),dV.$$

Likewise, a charge density can be integrated to obtain total charge, and other volumetric densities are treated in the same way.

Average value

If $V$ has finite nonzero volume, the average value of a scalar field over the region is

$$f_{\text{avg}}

\frac{1}{\operatorname{Vol}(V)} \iiint_V f,dV.$$

Volume integrals provide the natural language for adding quantities distributed continuously through space.