Triple Integrals

Integration of a function of three variables, w=f(x,y,z), over a three-dimensional region R in xyz-space is called a triple integral and is denoted

displaymath60

This page contains the following sections:

Triple Integrals in Box-Like Regions

Suppose that R is the box with a<=x<=b, c<=y<=d, and r<=z<=s.

The triple integral is given by

displaymath62

To compute the iterated integral on the left, one integrates with respect to z first, then y, then x. When one integrates with respect to one variable, all other variables are assumed to be constant. For a box-like region, the integral is independent of the order of integration, assuming f(x,y,z) is continuous. Hence, there are total of 6 ways to order the integrations. For example we can integrate with respect to x, then z, then y. In this case we have

displaymath64

Consider the following example:

displaymath66

The inner integral is

displaymath68

Integrating with respect to z, treating x and y as constants, we obtain

displaymath70

Note that z has completely disappeared from the expression on the right. The middle integral is with respect to y and x is treated as a constant. We have

displaymath72

Note that y has disappeared from the expression on the right. The outer integral is with respect to x. We have

displaymath74

You should verify that the same answer is obtained if the order of integration is changed.

Discussion and Applications

To get a better understanding of triple integrals let us consider the following example where the triple integral arises in the computation of mass. Suppose that that the region R in xyz-space corresponds to an object and f(x,y,z) is the density per unit volume at the point (x,y,z). If the density is constant, then the mass of the object is the product of the density and the volume of R. If the density varies with position, then we cannot apply this general formula.

We can compute the mass by slicing the R into a bunch of infinitesimal boxes. Consider the box between x and x+dx, y and y+dy, and z and z+dz. Here dx, dy, and dz are infinitesimals. In this small box the density is essentially constant and is equal to f(x,y,z). The mass of the small box is the product of density and volume. The volume of the box is dxdydz. Hence the mass of the the small box is f(x,y,z)dxdydz. The triple integral gives the total mass of the object and is equal to the sum of the masses of all the infinitesimal boxes in R.

Triple integrals also arise in computation of

Triple Integrals in General Regions

We would like to be able to integrate triple integrals for more general regions. General regions are classifed into three types. Suppose that the region R is such that g_1(x,y)<=z<=g_2(x,y), where (x,y) lies in region D in the xy plane, as shown in the figure:

The region D is the projection of R onto the xy plane. The triple integral is given by

displaymath76

The inner integral

displaymath78

is with respect to z. The result is a function of x and y. The remainder of the calculation

displaymath80

is a double integral over the region D in the xy plane.

If the region R is defined such that g_1(x,z)<=y<=g_2(x,z), (x,z) lies in region D in the xz plane, then

displaymath82

The inner integral is with respect to y. The remainder of the calculation involves a double integral over the region D in the xz plane.

Finally suppose that R is such that g_1(y,z)<=x<=g_2(y,z), where (y,z) lies in a region D in the yz plane, then

displaymath84

The inner integral is with respect x. The remainder of the calculation involves a double integral over the region D in the yz plane.

Example

Consider the triple integral

displaymath86

where R is the tetrahedral region bounded by the planes x=0, y=0, z=0 and x+y+z=2 (see figure below).

There are several ways to compute the integral. We can rewrite the equation of the plane x+y+z=2 as z=2-x-y. Note that 0<=z=<2-x-y. Hence, we have

displaymath88

The inner integral is (remember x and y are constants in this integration)

displaymath90

The projection of the region R onto the xy-plane is the triangle D shown in the figure below:

Hence, we are left with the double integral

displaymath92

We can also evaluate the double integral by integrating with respect to x first, then y. In this case

displaymath94

It can be shown that the double integral equals 2/3.


[Vector Calculus Home] [Math 254 Home] [Math 255 Home] [Notation] [References]

Copyright © 1996 Department of Mathematics, Oregon State University

If you have questions or comments, don't hestitate to contact us.