Calculus IV Fake Quiz 1 Fall 1999 10/15/99


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.