Page 218 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 9] MULTIPLE INTEGRALS 209

where the integral in braces is to be evaluated first (keeping x constant) and finally integrating with
respect to x from a to b. The result (4) indicates how a double integral can be evaluated by expressing it
in terms of two single integrals called iterated integrals.
The process of iterated integration is visually illustrated in Fig. 9-3a,b and further illustrated as
follows.

Fig. 9-3

The general idea, as demonstrated with respect to a given three-space region, is to establish a plane
section, integrate to determine its area, and then add up all the plane sections through an integration
with respect to the remaining variable. For example, choose a value of x (say, x ¼ x Þ. The intersection
0
of the plane x ¼ x with the solid establishes the plane section. In it z ¼ Fðx ; yÞ is the height function,
0
0
and if y ¼ f 1 ðxÞ and y ¼ f 2 ðxÞ (for all z) are the bounding cylindrical surfaces of the solid, then the width
ð
y 2
Fðx ; yÞ dy. Now establish slabs
0 0 0
is f 2 ðx Þ f 1 ðx Þ, i.e., y 2 y 1 . Thus, the area of the section is A ¼
y 1
A j x j , where for each interval x j ¼ x j x j 1 , there is an intermediate value x j . Then sum these to get
0
an approximation to the target volume. Adding the slabs and taking the limit yields
n ð ð
b
X y 2
V ¼ lim A j x j ¼ Fðx; yÞ dy dx
j¼1 y 1
n!1 a
In some cases the order of integration is dictated by the geometry. For example, if r is such that any
lines parallel to the x-axis meet the boundary of r in at most two points (as in Fig. 9-1), then the
equations of curves CAD and CBD can be written x ¼ g 1 ðyÞ and x ¼ g 2 ð yÞ respectively and we find
similarly
d g 2 ð yÞ
ðð ð ð
Fðx; yÞ dx dy
Fðx; yÞ dx dy ¼ ð5Þ
y¼c x¼g 1 ð yÞ
r
ð d ð g 2 ð yÞ
Fðx; yÞ dx dy
¼
y¼c x¼g 1 ð yÞ
If the double integral exists, (4)and (5) yield the same value. (See, however, Problem 9.21.) In writing a
double integral, either of the forms (4)or(5), whichever is appropriate, may be used. We call one form
an interchange of the order of integration with respect to the other form.