Page 230 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 9] MULTIPLE INTEGRALS 221

ðð ð
dx dy dz
ðaÞ Required volume ¼
r
2
2 6 4 x
ð ð ð
dz dy dx
¼
x¼0 y¼0 z¼0
2 6
ð ð
2
ð4 x Þ dy dx
¼
x¼0 y¼0
2
ð 6

ð4 x Þy dx
2
¼

x¼0 y¼0
ð 2
2
ð24 6x Þ dx ¼ 32
¼
x¼0
2
2 6 4 x
ð ð ð
dz dy dx ¼ 32 by part (a), since is constant. Then
(b) Total mass ¼
x¼0 y¼0 z¼0
ð 2 ð 6 ð 4 x 2
xdz dydx
Total moment about yz plane x¼0 y¼0 z¼0 24 3
Total mass ¼ Total mass ¼ 32 ¼ 4
x x ¼
Ð 2 Ð 6 Ð 4 x 2
Total moment about xz plane x¼0 y¼0 z¼0 ydz dydx 96
y y ¼ ¼ ¼ ¼ 3
Total mass Total mass 32
ð 2 ð 6 ð 4 x 2
zdz dydx
Total moment about xy plane x¼0 y¼0 z¼0 256 =5 8
Total mass Total mass 32 5
z z ¼ ¼ ¼ ¼
Thus, the centroid has coordinates ð3=4; 3; 8=5Þ.
y
Note that the value for y could have been predicted because of symmetry.
TRANSFORMATION OF TRIPLE INTEGRALS
9.13. Justify equation (11), Page 211, for changing variables in a triple integral.
By analogy with Problem 9.6, we construct a grid of curvilinear coordinate surfaces which subdivide the
region r into subregions, a typical one of which is r (see Fig. 9-17).

Fig. 9-17