Page 219 - Schaum's Outline of Theory and Problems of Advanced Calculus

210 MULTIPLE INTEGRALS [CHAP. 9

In case r is not of the type shown in the above figure, it can generally be subdivided into regions
r 1 ; r 2 ; .. . which are of this type. Then the double integral over r is found by taking the sum of the
double integrals over r 1 ; r 2 ; ... .

TRIPLE INTEGRALS
The above results are easily generalized to closed regions in three dimensions. For example,
consider a function Fðx; y; zÞ defined in a closed three-dimensional region r. Subdivide the region
into n subregions of volume V k , k ¼ 1; 2; ... ; n. Letting ð k ; k ; k Þ be some point in each subregion,
we form
n
X
lim Fð k ; k ; k Þ V k ð6Þ
n!1
k¼1
where the number n of subdivisions approaches infinity in such a way that the largest linear dimension of
each subregion approaches zero. If this limit exists, we denote it by
ðð ð
Fðx; y; zÞ dV ð7Þ
r
called the triple integral of Fðx; y; zÞ over r. The limit does exist if Fð; x; y; zÞ is continuous (or piecemeal
continuous) in r.
If we construct a grid consisting of planes parallel to the xy, yz, and xz planes, the region r is
subdivided into subregions which are rectangular parallelepipeds. In such case we can express the triple
integral over r given by (7)asan iterated integral of the form
ð b ð g 2 ðaÞ ð f 2 ðx;yÞ ð b ð g 2 ðxÞ ð f 2 ðx;yÞ
Fðx; y; zÞ dz dy dx
Fðx; y; zÞ dx dy dz ¼ ð8Þ
x¼a
x¼a y¼g 1 ðxÞ z¼f 1 ðx;yÞ y¼g 1 ðxÞ z¼f 1 ðx;yÞ
(where the innermost integral is to be evaluated first) or the sum of such integrals. The integration can
also be performed in any other order to give an equivalent result.
The iterated triple integral is a sequence of integrations; first from surface portion to surface portion,
then from curve segment to curve segment, and finally from point to point. (See Fig. 9-4.)
Extensions to higher dimensions are also possible.

Fig. 9-4