MULTIPLE INTEGRALS AND THEIR APPLICATIONS
411
44.32
44.33
Find the volume of a wedge cut from the cylinder 4x
2 + y
2 = a
2 by the planes z = 0 and z = my.
The base is the semidisk 9t bounded by the ellipse 4*
2 + y
2 = a
2
, y 2 0. Because of symmetry, we need
only double the first-octant volume. Thus,
Find I=((sin6dA, where &t is the region outside the circle r=l and inside the cardioid r=l + cosfl
(see Fig. 44-20).
For polar coordinates, recall that the factor r is introduced into the integrand via dA = rdr dO. By symmetry,
we can restrict the integration to the first quadrant and double the result.
Fig. 44-20
44.34
Use cylindrical coordinates to calculate the volume of a sphere of radius a.
44.35
44.36
44.37
Use polar coordinates to find the area of the region inside the circle x
2 + y
2 = 9 and to the right of the line
Fig. 44-21
Use polar coordinates to evaluate
The region of integration is the part of the unit disk in the first quadrant: 0:£0 Find the area of the region enclosed by the cardioid r = 1 + cos 0.
In cylindrical coordinates, the sphere with center (0, 0,0) is r
2 4- z
2 = a
2 . Calculate the volume in
the first octant and multiply it by 8. The base is the quarter disk 0 the standard formula.