Let *f*(*x*,*y*,*z*) be a function defined on a domain *R* in the three-dimensional space.
We divide the domain *R* with a network of lines parallel to the coordinate axes, and
we number the little boxes *A*_{1}, *A*_{2}, and so on. On each little
box *A*_{k} we choose a point
(*x*_{k}, *y*_{k}, *z*_{k}), as on Fig 7. The volume of such a little box is denoted
.

Define
.
If this sum
has a finite limit when we refine indefinitely the network, then this
limit is called the *triple integral* of *f* over *R* and is
denoted

If the function *f* is constant over *R* and *f*=1, then the triple integral

computes the volume of

In Calculus I, we learnt how to compute the volume of a solid of revolution; here we have a more general tool, which enables us to computes volumes for ``strange'' regions of the space.