Fig. 44-24
Fig. 44-25
44.41
Use integration in cylindrical coordinates to rind the volume of a right circular cone of radius a and heieht h
The base is the disk of radius a, given by r < a. For given r, the corresponding value of z on the cone is
determined by zl(a - r) = h/a (obtained by similar triangles: see Fie. 44-25.) Then
the standard
formula.
412
CHAPTER 44
Fig. 44-22
Fig. 44-23
44.39
44.38
Describe the planar region 01 whose area is given by the iterated integral
r - I - sin 0 is a cardioid, and r = 1 is the unit circle. Between 8 = TT and 0 = 2ir, 1< 1 - sin 6
and the cardioid is outside the circle. Therefore, 9). is the region outside the unit circle and inside the cardioid
(Fig. 44-22).
It suffices to double the area in the first quadrant, x
2 + y
2 = 9 becomes r = 3 in polar coordinates, and
is equivalent to r cos 0 = |. From Fig. 44-21, 0 < 0 s 77/3. So the area is
Evaluate the integral
using (a) rectangular coordinates and (b) polar coordinates.
(«)
(b) As indicated in Fig. 44-23, the region of integration lies under the semicircle
and above the line y = V3x (or 0 = ir/3).
Hence,
44.40
Find the volume of the solid cut out from the sphere x
2 + y
2 + z
2 < 4 by the cylinder x
2 + y
2 = 1 (see Fig
44-24).
It suffices to multiply by 8 the volume of the solid in the first octant. Use cvlindrical coordinate
The sphere is r +z =4 and the cylinder is r = l. Thus, we have