If F (x, y, z) is a function defined on a closed bounded region D in space (pretend that this region is a potato floating in space), then the integral of F over D maybe defined the following way. Partition a rectangular region containing D into rectangular cells by planes parallel to the coordinate planes. (Applying this concept to the potato, you first cut it into slices, then into french fries. The final cut makes rectangular cubes that resemble home fries.) Now, number these cells that lie inside D from 1 to n in some order. Pick a cell. This cell has side lengths Dx _k, Dy _k, and Dz _k and volume DV_k. Now choose a point (x _k, y _k, z _k) in each cell and form the sum

If F is continuous and the bounding surface of D is made of smooth surfaces joined along continuous curves, then as Dx _k, Dy _k, and Dz _k approach zero independently, the

Here are the properties of triple integrals.

1.

2.

3.

4.

5.

If D can be divided into a finite number of non-overlapping cells D₁, D₂, . . . D _n, then

If F is the constant function whose value is 1, then

Therefore,

FACT:

The volume of a closed, bounded region D in space is

EXAMPLE 1:

Write six different iterated triple integral for the volume of the rectangular solid in the first octant bounded by the coordinate planes and the planes x = 2, y = 3, and z = 4. Evaluate one of the integrals.

SOLUTION:

x goes between 0 and 2. y goes between 0 and 3, and z goes between 0 and 4.

EXAMPLE 2:

Write six different iterated triple integrals for the volume of the tetrahedron cut from the first octant by the plane 4x + 2y + 6z = 12. Evaluate one of the integrals.

SOLUTION:

EXAMPLE 3:

Find the volume of the wedge cut from the cylinder x ² + y ² = 1 by the planes z = - y and z = 0.

 SOLUTION:

EXAMPLE 4:

Find the volume of the region in the first octant bounded by the coordinate planes and the surface z = 4 - x ² - y.

SOLUTION:

I will integrate with respect to dz, then dy, and finally dx. z will go between 0 and 4 - x ² - y. y will go between 0 and 4 - x ², and x will go between 0 and 2.

EXAMPLE 5:

Find the average value of F (x, y, z) = x + y - z over the rectangular solid bounded by the coordinate planes and the planes x = 1, y = 1, and z = 2.

Here is the formula for the average value of a function.

The volume of the rectangular solid is (1)(1)(2) = 2.

EXAMPLE 6:

Evaluate the integral by changing the order of integration in an appropriate way.

As I look at the integrand of this triple integral, if I have to integrate with respect to x, I could not use a u-substitution, and integration by parts would be hairy. I think I will change the order of integration to by dy dx dz. By integrating with respect to y first, I will then have a u-substitution available when I integrate with respect to x next. Now, when x = 2y, then y = x/2. Here are the new integrals.

EXAMPLE 7:

Evaluate the integral by changing the order of integration in an appropriate way.

As I look at the integrand of this triple integral, integration with respect to x is fine, but the second integral is not straightforward. If I switch the order of dy and dz, then z will go between 0 and y ³, and y will go between 0 and 1.