,
which (with a straightforward u-substitution on the dr integral) works
out to 
.
This is not a real pop quiz. You do not have to turn
this in now or ever. This is strictly for your amusement, edification,
and spiritual betterment. Not to be taken internally.
1. [Ellis & Gulick p. 816] Find the volume of the solid region inside
the sphere x2+y2+z2=4, outside the cylinder
x2+y2=1, and above the xy plane.
Use cylindrical coordinates. It can be set up
as ,
which (with a straightforward u-substitution on the dr integral) works
out to .
2. [Ellis & Gulick p. 832] Find the volume of the solid region bounded
above by the circular paraboloid z=4(x2+y2), below
by the plane z=-2, and on the sides by the parabolic sheet y=x2
and the plane y=x.
The paraboloid would convert nicely to cylindrical
coordinates, but the plane and parabolic sheet wouldn't do we stay in rectangular.
The top view involves only the parabola y=x2 and line y=x, and
the region they bound has y=x2 below y=x and extends from x=0
to x=1, so we set up the integral
which works out to 71/105.