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CHAPTER 44
44.63
Find the volume within the cylinder r = 4 cos 0 bounded above by the sphere r
2 + z
2 = 16 and below by the
plane z = 0.
Refer to Fig. 44-34. Note that p = 2a cos 0 has the equivalent forms p
2 = 2ap cos , x
2 + y
2 + z
2 =
2az, x
2 + v
2 + (z - a)
2 = a
2 . Thus, it is the sphere with center at (0,0, a) and radius a. Then, V =
Integrate first with respect to z from z = 0 to
to
r = 4 cos 0,
and then with respect to 6 from
then with respect to r from r = 0
(See Fig. 44-35.)
Fig. 44-36
Fig. 44-35
44.64
44.65
44.66
44.67
Find the volume of the solid enclosed by the paraboloids
z = x
2 + y
2
(upward-opening) and
z = 36 - 3x2 - 8y2 (downward-opening).
The projection on the ry-plane is the circle x + y = 25. Use cylindrical coordinates.
Use a triple integral to find the volume V of the solid inside the cylinder x + y =25 and between the plane!
z = 2 and x + z = 8.
Evaluate
where ffl is the ball of radius a with center at the origin.
Use spherical coordinates.
Evaluate
where $1 is the tetrahedron bounded by the coordinate planes and the plane
x + 2y + z = 4 (see Fig. 44-36).
v can be integrated from 0 to 2. In the base triangle, for a given y, x runs from 0 to 4 - 2y. For given x
and y, z varies from 0 to 4 - x - 2y. Hence,