Calculus IV Problem Set 2

Calculus IV Problem Set 2 Fall 1998 10/19/98

1. [Stewart, p. 820, #34] If a function of one variable is continuous on an interval and has only one local maximum, then the maximum has to be an absolute maximum. But this is not true for functions of two variables. Show that the function

has exactly one critical point, and that f has a local maximum there that is not an absolute maximum. Then use a computer to produce a graph with a carefully chosen domain and viewpoint to see how this is possible.

2. Set up an iterated integral and use it to compute the volume of a rectangular box with sides of lengths a, b, and c.

3. Set up an iterated integral and use it to compute the volume of the solid with plane faces and vertices at (0, 0, 0), (a, 0, 0), (0, b, 0), and (0, 0, c).

4. Consider the truncated paraboloid bounded by the surfaces z = x² + y² and z = a, for some positive constant a. Express its volume as an iterated integral, and find its volume.

5. Compare the volume of the truncated paraboloid from problem 4 to the volumes of the cone inscribed in it and the cylinder circumscribed around it.

6. Looking at a graph of the function , we can see an infinite number of "ripples" (think of each "ripple" as the solid region above the xy plane) spreading outward from the origin. Find a formula for the volume of the n^th ripple, and show why it works.