Page 232 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 9] MULTIPLE INTEGRALS 223

2
2
9.17. Find the volume of the region above the xy plane bounded by the paraboloid z ¼ x þ y and the
2
2
2
cylinder x þ y ¼ a .
The volume is most easily found by using cylindrical coordinates. In these coordinates the equations
2
for the paraboloid and cylinder are respectively z ¼ and ¼ a. Then
Required volume ¼ 4 times volume shown in Fig. 9-18

ð =2 ð a ð 2
¼ 4 dz d d
¼0 ¼0 z¼0
ð =2 ð a
3
¼ 4 d d
¼0 ¼0
=2

ð 4 a
4
¼ 4 d ¼ a
hi¼0 4 ¼0 2

Fig. 9-18

2
The integration with respect to z (keeping and constant) from z ¼ 0to z ¼ corresponds to
summing the cubical volumes (indicated by dVÞ in a vertical column extending from the xy plane to the
paraboloid. The subsequent integration with respect to (keeping constant) from ¼ 0to ¼ a
corresponds to addition of volumes of all columns in the wedge-shaped region. Finally, integration with
respect to corresponds to adding volumes of all such wedge-shaped regions.
The integration can also be performed in other orders to yield the same result.
We can also set up the integral by determining the region r in ; ; z space into which r is mapped by
0
the cylindrical coordinate transformation.

9.18. (a) Find the moment of inertia about the z-axis of the region in Problem 9.17, assuming that the
density is the constant . (b) Find the radius of gyration.
(a) The moment of inertia about the z-axis is

ð =2 ð a ð 2
2
I z ¼ 4 dz d d
0 ¼0 z¼0
6

ð =2 ð a ð =2 6 a a
5
¼ 4 d d ¼ 4 d ¼
¼0 ¼0 ¼0 6 ¼0 3