In this section we are going to relate surface integrals to
triple integrals. We will do this with
the Divergence Theorem.
Let E be a
simple solid region and S is the
boundary surface of E with positive
Let’s see an example of how to use this theorem.
Example 1 Use
the divergence theorem to evaluate
Let’s start this off with a sketch of the surface.
The region E for
the triple integral is then the region enclosed by these surfaces. Note that cylindrical coordinates would be
a perfect coordinate system for this region.
If we do that here are the limits for the ranges.
We’ll also need the divergence of the vector field so
let’s get that.
The integral is then,