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In this section we are going to relate surface integrals to
triple integrals. We will do this with
the Divergence Theorem.

Divergence Theorem

Let E be a
simple solid region and S is the
boundary surface of E with positive
orientation. Let be a vector field whose components have
continuous first order partial derivatives.
Then,

Let’s see an example of how to use this theorem.

Example 1 Use
the divergence theorem to evaluate where and the surface consists of the three
surfaces, ,
on the top, ,
on the sides and on the bottom.

Solution

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.