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CHAPTER 44
44.55
Find the surface area S of the part of the sphere x
2 + y
2 + z
2 = 4z inside the paraboloid z = x
2 + y
2 .
The region 91 under the spherical cap (Fig. 44-33) is obtained by finding the intersection of x
2 + y
2 + z
2 = 4z
and z = x
1 + v
2 - This gives z(z-3) = 0. Hence, the paraboloid cuts the sphere when z = 3. and 9?
is the disk x
2 + y
2 < 3.
Similarly,
Hence,
Therefore,
Fig. 44-33
44.56
Find the surface area of the part of the sphere x
2 + y
2 + z
2 = 25 between the planes z =2 and z = 4.
The surface lies above the ring-shaped region 92:
3 s r s VJT.
since r
2 + z
2 = 25. Hence,
44.57
Find the surface area of a sphere of radius a.
Consider the upper hemisphere of the sphere x
2 + y
2 + z
2 = a
2 . It projects down onto the disk ffi of radius a
whose center is at the origin. Hence, the surface area of the entire sphere is
44.58
Find the surface area of a cone of height h and radius of base b.
Consider the cone
or b
2 z
2 = h
2 x
2 + h
2 y
2
(see Fig. 44-30). The portion of the
cone under
z = h
projects onto the interior 91 of the circle
r = b
in the jy-plane. Then