Fig. 44-32
MULTIPLE INTEGRALS AND THEIR APPLICATIONS
415
44.52
Find the average distance p from the center of a ball 3& of radius a to all other points of the ball.
44.53
Find the area 5 of the part of the plane x + 2y + z = 4 which lies inside the cylinder x
2 + y
2 = \.
Recall (Fig. 44-31) the relation
between an element of area dS of a
surface
and its projection dA in the xy-plane. Thus, the formula for S is
where, here, $ is the disk x
2 + y
2 <1.
Hence,
Fig. 44-31
44.54
Find the surface area 5 of the part of the sphere x
2 + y
2 + z
2 = 36 inside the cylinder x
2 + y
2 = 6y and
above the xy-plan'e.
x
2 + y
2 = 6y is equivalent to x
2 + (y - 3)
2 = 9. So the cylinder has axis x = 0, y = 3, and radius 3.
The base 3? is the circle x
2 + (y -3)
2 = 9. (See Fig. 44-32.)
Hence,
Similarly,
Thus,
Therefore,
Now use polar coordinates. The circle x
2 + y
2 = 6y is
equivalent to r = 6 sin 0. x
2 + y
2 + z
2 = 36 is equivalent to r
2 + z
2 = 36, or z
2 = 36 - r
2
. Hence, S =
Hence,