Calculus

Surface Integrals

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The Divergence Theorem

Solved Problems

Example 3.

Use the Divergence Theorem to evaluate the surface integral of the vector field where is the surface of a solid bounded by the cone and the plane (Figure ).

Solution.

The solid is sketched in Figure

The surface of a solid bounded by the cone x^2+y^2−z^2=0 and the plane z=1
Figure 2.

Applying the Divergence Theorem, we can write:

By changing to cylindrical coordinates, we have

Example 4.

Using the Divergence Theorem calculate the surface integral of the vector field where is the surface of tetrahedron with vertices (Figure ).

Solution.

Tetrahedron with vertices O(0,0,0), A(1,0,0), B(0,1,0), C(0,0,1)
Figure 3.

By Divergence Theorem,

Find the given triple integral. The equation of the straight line has the form:

The equation of the plane is

So the integral becomes:

Example 5.

Calculate the surface integral of the vector field where is the surface of the rectangular box bounded by the planes (Figure ).

Solution.

A rectangular box
Figure 4.

Using the Divergence Theorem, we can write:

Example 6.

Find the surface integral

where is the outer surface of the pyramid

(see Figure ).

Solution.

A triangular pyramid
Figure 5.
The base of triangular pyramid
Figure 6.

Using the Divergence Theorem, we can write the initial surface integral as

Calculate the triple integral. The region of integration in the -plane is shown in Figure By setting we find:

Hence, the region can be represented in the form:

We rewrite the inequality in terms of

Then the triple integral becomes

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