Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MULTIPLE INTEGRALS

z

x

y

C

R

T

S P Q

u=c 1

v=c 2

w=c 3

Figure 6.12 A three-dimensional region of integrationR, showing an el-

ement of volume inu, v, wcoordinates formed by the coordinate surfaces

u=constant,v=constant,w=constant.

evaluate the integrals over the inner and outer circles respectively, and we find

π

(

1 −e−a

2 )

<I^2 (a)<π

(

1 −e−^2 a

2 )

.

Taking the limita→∞, we findI^2 (a)→π. ThereforeI=

√

π, as we found previ-

ously. Substitutingx=

√

αyshows that the corresponding integral of exp(−αx^2 )

has the value

√

π/α. We use this result in the discussion of the normal distribution

in chapter 30.

6.4.3 Change of variables in triple integrals

A change of variable in a triple integral follows the same general lines as that for

a double integral. Suppose we wish to change variables fromx, y, ztou, v, w.

In thex, y, zcoordinates the element of volume is a cuboid of sidesdx, dy, dz

and volumedVxy z=dx dy dz. If, however, we divide up the total volume into

infinitesimal elements by constructing a grid formed from the coordinate surfaces

u=constant,v=constantandw= constant, then the element of volumedVuvw

in the new coordinates will have the shape of a parallelepiped whose faces are the

coordinate surfaces and whose edges are the curves formed by the intersections of

these surfaces (see figure 6.12). Along the line elementPQthe coordinatesvand