Triple Integrals

Triple integrals Consider

defined on a closed and bounded domain

. Then

can be contained in a cuboid

. We divide the cuboid into small sub-cuboids by planes parallel to xy, yz, and xz planes. Choose the small sub-cuboids that are contained in

and label them from

. For the

sub-cuboid, we denote its side lengths to be

, and

. Then its volume

. Moreover, we select an arbitrary point

in the sub-cuboid. The triple integral of

over

is defined to be the limit of the following "Riemann sum":

The diagram shown below is the limiting procedure of cutting the domain

into smaller and smaller sub-cuboids.

Physical interpretation of triple integral Suppose

is a 3D solid and its density varies at different points of the solid according to

. Then

Total mass of

. In particular, if

, i.e. the solid has "uniform density" of 1, we can regard the triple integral

as the volume of the solid. Suppose

, i.e. a cuboid. We can calculate the triple integral using the following version of Fubini's theorem: Fubini's theorem: Suppose

and

. Then we have

This integral is also equal to any of the other five possible orderings for the iterated triple integral. Example: Find the value of

, where

. Answer:

Exercise: Find the mass of the cuboid with density .

Triple integrals over a general domain Suppose

is defined on a domain

, which is solid bounded by the graph of

from below and by the graph of

from above over a region

on the xy plane. In other words,

for all

. This kind of domains is called a simple xy-solid. We can reduce the triple integral over a simple xy-solid to the double integral as follows:

Moreover, if

is a Type I region bounded from below by the graph of

and from above by the graph of

over

, we can express the triple integral as the following iterated integral:

We have similar results for simple xz-solid and simple yz-solid:

is a simple xz-solid if it is bounded by the surfaces

and

over the region

on xz plane and

for all

. Then the triple integral over

is as follows:

(The definition of simple yz-solid and its analogous result are left as exercise.)

Example: Let

be the wedge in the first octant i.e. the set of points in

such that x,y,z coordinates are non-negative, that is cut from the cylindrical solid

by the planes

and

. Find the value of the triple integral

. Answer: The domain

can be regarded as a simple xy-solid such that it is bounded by the cylinder

from above and the plane

from below over a triangular region

as shown in the applet below. Notice that

(Note: Positive square root is used because

is non-negative) Then, we have

Moreover,

can be regarded as a Type II region bounded by

and

. Therefore, we can express the triple integral as the following iterated integral:

Exercise: Express the triple integral in the above example as an iterated integral by regarding as a simple yz-solid. You are not required to compute the iterated integral.

Example: Find the volume of the solid

bounded by the two paraboloids

and

. Answer: Volume of

can be computed by evaluating the triple integral

. And

can be regarded as a simple xz-solid bounded by

and

. In order to find the region

on which the solid is bounded by these two surfaces, we need to find the projection of the intersection of these two surfaces to the xz plane. First, we combine the two equations of the surfaces. We have

i.e., the projection of the intersection onto xz plane is a ellipse, as shown in the applet below. Therefore,

is the region bounded by the ellipse, which can be regarded as a Type I region bounded by

and

for

. Hence, we can write the triple integral as the following iterated integral:

(Note: Use substitution

)

Changing the order of integration Sometimes it is necessary to change the order of integration in an iterated integral so that the it can be evaluated more easily. Example: Suppose

is a 3D solid such that

Describe the solid

and then change the order of integration in the iterated integral to evaluate the triple integral. Answer: Since the innermost integral is with respect to

, the solid

is a simple yz-solid bounded by

and

. Moreover, the integrals with respect to

and

suggest that the simple yz-solid is over the region

on yz plane, where

is the region bounded by

and

for

. The applet below shows that

is a tetrahedron bounded by four planes:

, and

. We can view

as a simple xz-solid bounded by

and

over the triangular region bounded by

and

for

. Therefore, we can write the iterated integral in a different order as follows:

Exercise: Evaluate the following iterated integral by reiterating it in a different order. . (Note: You will need to make a good sketch of the region.)

Triple Integrals

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