Page 224 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 9] MULTIPLE INTEGRALS 215

ffiffiffiffiffiffiffiffi
p
"# 2 x 2
p ffiffiffiffiffiffiffiffi
1
b
y
ð ð f 2 ð ð 2 x 2 ð 1 2
xdx
ðbÞ M ¼ dy dx ¼ yx dy dx ¼
a f 1 0 x 2 0 2
x 2
" # 1
1 1 2 4 x x x 7
ð 2 4 6
0 2 2 8 12 24
¼ xð2 x x Þ dx ¼ ¼
0
(c) The coordinates of the center of mass are defined to be
b
b
1 ð ð f 2 ðxÞ 1 ð ð f 2 ðxÞ
x dy dx and y dy dx
M a f 1 ðxÞ M a f 1 ðxÞ
x x ¼ y y ¼
where
b f 2 ðxÞ
ð ð
dy dx
M ¼
a f 1 ðxÞ
Thus,
ffiffiffiffiffiffiffiffi
p
"# 2 x 2
p ffiffiffiffiffiffiffiffi
1
ð ð 2 x 2 ð 1 2 ð 1
4
2
x x 2 y x 2 1 ½2 x x Š dx
M x ¼ xxy dy dx ¼ 2 dx ¼ 2
0 x 2 0 0
x 2
" 3 5 7 # 1
x x x 1 1 1 17
3 10 14 3 10 14 105
¼ ¼ ¼
0
p ffiffiffiffiffiffiffiffi
1 2 x 13 2
ð ð 2 p ffiffiffi
y þ 4
M y ¼ yx dy dx ¼
0 x 2 120 15
2 2 2
9.4. Find the volume of the region common to the intersecting cylinders x þ y ¼ a and
2 2 2
x þ z ¼ a .
Required volume ¼ 8 times volume of region shown in Fig. 9-9
p ffiffiffiffiffiffiffiffiffiffi
2
ð a ð a x 2
¼ 8 zdy dx
x¼0 y¼0
p ffiffiffiffiffiffiffiffiffiffi
ð a ð 2 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a x p
2
2
¼ 8 a x dy dx
x¼0 y¼0
a 16a
ð 3
2
2
¼ 8 ða x Þ dx ¼
x¼0 3
As an aid in setting up this integral, note that zdy dx corresponds to the volume of a column such as
shown darkly shaded in the figure. Keeping x constant and integrating with respect to y from y ¼ 0to
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p 2 2
a x corresponds to adding the volumes of all such columns in a slab parallel to the yz plane, thus
y ¼
giving the volume of this slab. Finally, integrating with respect to x from x ¼ 0to x ¼ a corresponds to
adding the volumes of all such slabs in the region, thus giving the required volume.
9.5. Find the volume of the region bounded by
z ¼ x þ y; z ¼ 6; x ¼ 0; y ¼ 0; z ¼ 0