In mathematics — in particular, in multivariable calculus — the term volume integral is used in two different, albeit related, ways. In some contexts, it means a triple integral within a region D of the function f(x,y,z) = 1 usually written as:

which gives the volume of D.

In other contexts, it means the triple integral of a function f(x,y,z)

Examples

gives the volume of the unit sphere.

Analogous to this, if ρ = ρ(x,y,z) describes the density of D, then the total mass of the body D is given by:

Divergence theorem

By using the the divergence theorem one can find an interesting expression for the volume of a body. Let D be a region in R³, and S be its boundary. Conisder the vector field

F(x,y,z) = (x,0,0)

for which we have div F = 1. Conisder a parametrization of S with the normal pointing out of the body. Then, by the divergence theorem, the volume of D is given by

External link

• MathWorld article on volume integrals