Each problem is worth 10 points. Show all work for full credit. Please circle all answers and keep your work as legible as possible. May not be safe for use as drinking water.

1. Calculate the double Riemann sum of the function f(x,y) = 3x + y² for the partition of R=[0,1]×[-1,1] given by x=½ and y=0, using (x_ij^*,y_ij^*) = the midpoint of R_ij.
 
 

2. The Oklahoma legislature has decided that instead of giving graduate teaching assistants a long-deserved and much-needed pay increase, they're going to use the funds to tear down the Physical Sciences building and replace it with an even goofier structure with a base shaped like the rectangle [-30,30]×[-40,40] in the xy-plane and with the top shaped like the portion of the paraboloid z=2500-x²-y². Set up a double or triple integral for the volume of this new building.
 
 
 

3. Set up a double integral for the area of the region in the xy plane which lies inside a circle with radius 3 and outside the cardioid r=1+cos .
 
 

4. Archeologists find a coprolite shaped like the solid bounded by the paraboloid z=6-x²-y² and the plane z=0. As part of their analysis they drill out a hole down the center (i.e., along the z axis) of radius 1. Set up a multiple integral for the volume of the remaining solid.
 
 

5. For some really, really important reason you need to make a change of variable in a double integral, where the desperately needed transformation is given by x=3u-6v, y=-2u+4v. Find the Jacobian for this transformation.
 
 

6. Find the exact value of .
 
 

7. Barbie is a Calc IV student from California. Barbie says "Multiple integrals are hard. I can never figure out what to put for the limit thingies. Like, it's always different and so confusing, but then I think I got a plan. I, like, started just always doing in the first quarter-thingy, you know? And then I just multiply it by four if the picture is all above the, you know, the middle-thingy, or by eight if some of it is below the middle-thingy. That makes it a lot less stress."

Clearly explain, in a way that Barbie can understand, why what she's doing might be valid or invalid. Be sure to include some specifics about when it will or will not work.
 
 

8. A partially decomposed moose carcass happens to be shaped exactly like the region bounded by z=4-x² and the planes x+y=6, y-x=6, y=0, and z=0. Find the volume of the carcass.
 
 

9. Find the centroid of the portion of a sphere of radius 1 which lies in the first octant.
 
 

10. Find the surface area of the portion of the plane z=ax which lies inside the cylinder x²+y²=R².
 
 

Extra Credit:

The familiar formulas V=r²h and S=2r²+2rh for the volume and surface area of a cylinder are usually applied only to right circular cylinders, i.e. the ones like tin cans, where the cross sections are circles and the ends are at right angles to the side or axis. Do these formulas also work for circular cylinders which are not right, but instead have ends which are still parallel to each other, but not necessarily at right angles to the side or axis?
 

Conjectures are okay, but those supported by some kind of work are better.