414
CHAPTER 44
Fig. 44-29
44.47
Show that
Then
Hence,
Let
The region of integration is the entire first quadrant. Change
to polar coordinates.
(The rather loose reasoning
in this computation can be made rigorous.)
44.48
Evaluate
by Problem
Let
44.47.
Then
44.49
Evaluate
44.50
44.51
Use spherical coordinates to find the volume of a sphere of radius a.
Consider
Use integration by parts. Let
Then
Hence,
by Problem 44.47.
In spherical coordinates a sphere of radius a is characterized by 0 Recall that the volume element is given by dV= p
2 sin dp dd d Use spherical coordinates to find the volume of a right circular cone of height h and radius of base b.
For the orientation shown in Fig. 44-30, the points of the cone satisfy 0<0s2ir, Os<^> 0 -s p < h sec A. Thus,
Fig. 44-30
Since