Page 229 - Schaum's Outline of Theory and Problems of Advanced Calculus

220 MULTIPLE INTEGRALS [CHAP. 9

(c) The triple integral can be expressed as the iterated integral
ð a ð a x ð a x y
2
2
2
ðx þ y þ z Þ dz dy dx
x¼0 y¼0 z¼0

ð a ð a x 3 a x y
2 2 z
dy dx
¼ x z þ y z þ
x¼0 y¼0 3 z¼0
( )
ð a ð a x 3
2 2 2 3 ða x yÞ
¼ x ða xÞ x y þða xÞy y þ 3 dy dx
x¼0 y¼0
a x y ða xÞy y
ð 2 2 3 4 4 a x

2 ða x yÞ
dx
¼ x ða xÞy þ
x¼0 2 3 4 12 y¼0
( )
a
ð 2 2 4 4 4
2 2 x ða xÞ ða xÞ ða xÞ ða xÞ
¼ x ða xÞ þ þ dx
0 2 3 4 12
( )
ð a 2 2 4 a 5
x ða xÞ ða xÞ
¼ þ dx ¼
0 2 6 20
The integration with respect to z (keeping x and y constant) from z ¼ 0to z ¼ a x y corre-
sponds to summing the polar moments of inertia (or masses) corresponding to each cube in a vertical
column. The subsequent integration with respect to y from y ¼ 0to y ¼ a x (keeping x constant)
corresponds to addition of contributions from all vertical columns contained in a slab parallel to the yz
plane. Finally, integration with respect to x from x ¼ 0to x ¼ a adds up contributions from all slabs
parallel to the yz plane.
Although the above integration has been accomplished in the order z; y; x, any other order is
clearly possible and the final answer should be the same.
9.12. Find the (a) volume and (b) centroid of the region r bounded by the parabolic cylinder
2
z ¼ 4 x and the planes x ¼ 0, y ¼ 0, y ¼ 6, z ¼ 0 assuming the density to be a constant .
The region r is shown in Fig. 9-16.

Fig. 9-16