Page 216 - Schaum's Outline of Theory and Problems of Advanced Calculus

Multiple Integrals

Much of the procedure for double and triple integrals may be
thought of as a reversal of partial differentiation and otherwise is
analogous to that for single integrals. However, one complexity
that must be addressed relates to the domain of definition. With
single integrals, the functions of one variable were defined on
intervals of real numbers. Thus, the integrals only depended on
the properties of the functions. The integrands of double and
triple integrals are functions of two and three variables, respec-
tively, and as such are defined on two- and three-dimensional
regions. These regions have a flexibility in shape not possible
in the single-variable cases. For example, with functions of two
variables, and the corresponding double integrals, rectangular Fig. 9-1
regions, a @ x @ b, c @ y @ d are common. However, in
many problems the domains are regions bound above and below by segments of plane curves. In
the case of functions of three variables, and the corresponding triple integrals other than the regions
a @ x @ b; c @ y @ d; e @ z @ f , there are those bound above and below by portions of surfaces. In
very special cases, double and triple integrals can be directly evaluated. However, the systematic
technique of iterated integration is the usual procedure. It is here that the reversal of partial differentia-
tion comes into play.
Definitions of double and triple integrals are given below. Also, the method of iterated integration
is described.

DOUBLE INTEGRALS
Let Fðx; yÞ be defined in a closed region r of the xy plane (see Fig. 9-1). Subdivide r into n
subregions r k of area A k , k ¼ 1; 2; ... ; n. Let ð k ; k Þ be some point of A k . Form the sum
n
X
Fð k ; k Þ A k ð1Þ
k¼1
Consider
n
X
lim Fð k ; k Þ A k ð2Þ
n!1
k¼1
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