Each problem is worth 10 points. Show all work for full credit. Please circle all answers and keep your work as legible as possible. This test has no adverse ecological impact on threatened species.

1. Use a double Riemann sum to approximate the value of  for R={(x,y) | 0 x 2, 3 y 7} using the partition lines x=1 and y=5 and letting {x_ij*, y_ij*) be the center of R_ij.
 
 

2. Write an iterated integral (in your choice of coordinate systems) with appropriate limits to integrate a function f over each of the following regions:
(a) The solid region between a sphere (centered at the origin) of radius 3 and a sphere (centered at the origin) of radius 5 below the plane z=0.

(b) A wedge of cheddar cheese sitting on the xy plane in the first octant with radius 3 and height 2, with one face on the plane x=0 and the angle between the faces /6.
 
 

3. Suppose you're walking down the street one day and you find an arbitrarily thin, flat sheet of some rigid slug-colored substance shaped like the inner loop of the limaçon r=1+2sin . Set up a double integral for the area of this object.
 
 

4. Find the surface area of the portion above the xy plane of the cone z²=x²+y² which lies within the cylinder x²+y²=9.
 
 

5. Compute the integral 
 
 

6. A forest next to a road has the shape shown below. The population density of marmots in the forest is given by D(x,y)=10-2y marmots per square mile. Find the total marmot population in the forest.



 
 
 
 

7. Biff says "Uh, so my Calc professor said I was wrong on this thing on the test. What happened was I got a negative number for the volume of this thing, so I just took off the negative sign and circled that. I mean, if you get a negative for volume it must have just been the opposite of the right answer, right? But she said, uh, like this confusing stuff that I didn't understand, and said that wasn't right. But she said I didn't screw up working it out, so I think she just hates me because I found a shortcut."

Either (if you think Biff's method is right) explain to Biff's professor why what he did is valid, or (if you think Biff's professor is right) explain to Biff what goes wrong with his "shortcut" method.
 
 

8. Use the transformation x=au, y=bv to compute  for the region R bounded by the ellipse .
 
 

9. Space aliens have just landed and given you a lamina of uniform density shaped like a right triangle and explained that you should think of it as lying in the first quadrant with the leg of length a along the positive x axis and the leg of length b lying along the positive y axis. They demand that you find either the x or y coordinate of the center of mass (your choice) or they'll exterminate life on Earth.
 
 

10. Find the volume of the region above the plane z=0 and below both the surface z=4-x² and the surface z=4-y².

 

Extra Credit (5 points possible):

Generalize problem 10 to the situation where the two parabolic cylinders are z=a-x² and z=b-y² [if you can't do the whole thing, at least find the dimensions of the projection in the xy plane, i.e. the top view].