Page 234 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 9] MULTIPLE INTEGRALS 225

(b)Letting ¼ ,the volume of the sphere thus obtained is
2 a 3 4 3
ð1 cos Þ¼ a
3 3

Find the centroid of the region in Problem 9.19.
9.20. ðaÞ
(b) Use the result in (a)to find the centroid of a hemisphere.
y
x
x
z
y
(a) The centroid ð x; y; zÞ is, due to symmetry, given by x ¼ y ¼ 0 and
ÐÐÐ
Total moment about xy plane z dV
¼ ÐÐÐ
z z ¼
Total mass dV
Since z ¼ r cos and is constant the numerator is
=2 a =2
ð ð ð ð ð 4 a

2
4 r cos r sin dr d d ¼ 4 r sin cos d d
¼0 ¼0 r¼0 ¼0 ¼0 4 r¼0
=2
ð ð
¼ a 4 sin cos d d
¼0 ¼0
4
2
ð =2 2 a sin
¼ a 4 sin d ¼
¼0 2 ¼0 4
3
2
The denominator, obtained by multiplying the result of Problem 9.19(a)by ,is a ð1 cos Þ.
3
Then
4
2
1 a sin 3
4 ¼ að1 þ cos Þ:
2 3 8
z z ¼
3 a ð1 cos Þ
3
z
(b)Letting ¼ =2; z ¼ a.
8
MISCELLANEOUS PROBLEMS
1 ð 1 x y 1 1 ð 1 x y 1
ð ð
9.21. Prove that (a) 3 dy dx ¼ ,(b) 3 dx dy ¼ .
0 0 ðx þ yÞ 2 0 0 ðx þ yÞ 2
1 1 x y 1 1
ð ð ð ð
dy dx
2x ðx þ yÞ
3 3
ðaÞ dy dx ¼
0 0 ðx þ yÞ 0 0 ðx þ yÞ
1
1
ð ð 2x 1
dy dx
3 2
¼
0 0 ðx þ yÞ ðx þ yÞ
1
ð 1
x 1
dx
2 x þ y
¼ þ
0 ðx þ yÞ y¼0
1 dx 1 1
ð 1

2 x þ 1 2
¼ ¼ ¼
0 ðx þ 1Þ 0
1
ð ð 1 y x 1
(b) This follows at once on formally interchanging x and y in (a)to obtain 3 dx dy ¼ 2 and
then multiplying both sides by 1. 0 0 ðx þ yÞ
This example shows that interchange in order of integration may not always produce equal results.
Asufficient condition under which the order may be interchanged is that the double integral over the
ðð
x y
corresponding region exists. In this case dx dy, where r is the region
3
ðx þ yÞ
r
0 @ x @ 1; 0 @ y @ 1fails to exist because of the discontinuity of the integrand at the origin. The
integral is actually an improper double integral (see Chapter 12).
x t x
ð ð ð
9.22. Prove that FðuÞ du dt ¼ ðx uÞFðuÞ du.
0 0 0