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 A1, A2, and so on. On each little
box Ak we choose a point
(xk, yk, zk), as on Fig 7. The volume of such a little box is denoted
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
If the function f is constant over R and f=1, then the triple integral
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.