Calculus IV Exam 2 Fall 1998 10/22/98

Each problem is worth 10 points. Show all work for full credit. Please circle all answers and keep your work as legible as possible. It's neither optimistic nor pessimistic, really.

1. A field biologist determines that the population density of dwarf carnivorous moose in a particular region is roughly given by dwarf carnivorous moose per square mile. Approximate the number of dwarf carnivorous moose in the region using the partition given by the lines x = 10 and y = 20, with midpoints of each rectangle as representatives (x_ij^*, y_ij^*).

2. If the Earth is taken to be a sphere 3963 miles in radius and the atmosphere is 600 miles thick, write a triple integral in spherical coordinates that would produce the total amount of ozone above the northern hemisphere given a function for the density of ozone per cubic mile.

3. Show that the Jacobian for the transformation to spherical coordinates (, , ) is .

4. Evaluate the integral .

5. Set up limits of integration for if the region E is bounded by x = 0, y² + z² = 9, and x = 10 - z. You need not work out the integral.

6. Evaluate the integral

7. Show that the center of mass of a beautiful purple disk of any radius, centered at the origin and with constant density, is at the origin.

8.Show that the area of the part of the plane z = ax + by + c that projects onto a region D in the xy-plane with area A(D) is .

9. Buffy is a calculus student at Oklahoma State. She says "Like, our professor told us that when you use that Fubini thingy, like, all the little limits switch around, y'know? But, like, y'know, I figured out that it's the same when you do as when you do , so, I mean, I guess it's okay to just switch, like, the dz and the dy, or whatever, y'know?"

Is Buffy right, or are there corrections or limitations that should be made to her statement?

10. A truncated paraboloid is formed between the surface z = x² + y² and the plane z = a for some positive constant a. To what depth should the paraboloid be filled with water in order that the water have exactly half the volume of the whole solid? (Yes, you need to work out the formula for the volume of the paraboloid even if you remember it from the problem set.)

Extra Credit (5 points possible):

Show why the Jacobian of the transformation x = u(x,y), y = v(x,y) is .