Mathematical Methods for Physics and Engineering : A Comprehensive Guide

6.4 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS

y

x

b

dx a

dM=σb dx

Figure 6.9 A uniform rectangular lamina of massMwith sidesaandbcan

be divided into vertical strips.

6.3.5 Mean values of functions

In chapter 2 we discussed average values for functions of a single variable. This

is easily extended to functions of several variables. Let us consider, for example,

a functionf(x, y) defined in some regionRof thexy-plane. Then the average

value ̄fof the function is given by

̄f

∫

R

dA=

∫

R

f(x, y)dA. (6.10)

This definition is easily extended to three (and higher) dimensions; if a function

f(x, y, z) is defined in some three-dimensional region of spaceRthen the average

value ̄fof the function is given by

f ̄

∫

R

dV=

∫

R

f(x, y, z)dV. (6.11)

A tetrahedron is bounded by the three coordinate surfaces and the planex/a+y/b+z/c=

1 and has densityρ(x, y, z)=ρ 0 (1 +x/a). Find the average value of the density.

From (6.11), the average value of the density is given by

̄ρ

∫

R

dV=

∫

R

ρ(x, y, z)dV.

Now the integral on the LHS is just the volume of the tetrahedron, which we found in
subsection 6.3.1 to beV=^16 abc, and the integral on the RHS is its massM= 245 abcρ 0 ,
calculated in subsection 6.3.2. Therefore ̄ρ=M/V=^54 ρ 0 .

6.4 Change of variables in multiple integrals

It often happens that, either because of the form of the integrand involved or

because of the boundary shape of the region of integration, it is desirable to