Each problem is worth 10 points. Be sure to show all work for full credit. Please circle all answers and keep your work as legible as possible. Not intended for use as protective headgear.
 

1. Approximate the integral  for the partition of R = {(x,y)|0x2, 0y1} given by the lines x=1 and y=½, taking (x_i*, y_i*) to be the center of each subrectangle.
 
 

2. Set up an integral for M_x, the moment about the x axis, of a lamina with density  and shaped like the left half of a circle with radius 10,

a) In rectangular coordinates

b) In polar coordinates
 
 

3. An artist plans to build a large abstract ice sculpture at the North pole representing the essence of polar bears. It will be shaped like the region bounded by the plane z=0, the plane y=0, the cylinder x² + z² = 25, and the plane x + y = 10. Set up an integral for the volume of the sculpture.
 
 

4. In computing the area of an ellipse it can be convenient to make the transformation x = a r cos, y = b r sin. Find the Jacobean for this transformation.
 
 

5. A group of happy little bunnies lives in a forest. If the total bunny population is given by the integral ,

a) What function (x,y) gives the population density of bunnies at a point in the forest?

b) What is the total bunny population in the forest, to the nearest bunny? [Hint: It may help to
reverse the order of integration].
 
 

6. The OU Math Club is considering producing and marketing small crimson Math Club beanies shaped like the part of the sphere x² + y² + z² = 9 that lies within the cylinder x² + y² = 4. Find the surface area of such a beanie.
 
 

7. Evaluate the integral .
 
 

8. Biff is a calculus student at O.S.U. who's having some trouble with integrating in spherical coordinates. He's just set up an integral  for the volume of a certain region. Is there anything you can suggest Biff might need to correct about his work?
 
 

9. Let s be any arc of the unit circle lying entirely in the first quadrant. Let A be the area of the region lying below s and above the x-axis and let B be the area of the region lying to the right of the y-axis and to the left of s. Prove that A + B depends only on the arc length, and not on the position of s. [Hints: Sketch yourself a picture and label the points at the beginning and end of the arc s as (cos ₁, sin ₁) and (cos ₂, sin ₂). Setting up integrals for A and B is worth half the points. In working them out, the formulas  and  might be helpful.]
 
 

10. The region created by removing a smaller square from one corner of a larger square is called a square gnommon. If a lamina with uniform density and shaped like a square gnommon is created by removing a square with side length a from a square with side length b, where will the center of mass of the lamina lie?
 
 

Extra Credit (5 points possible):

Depending on the values of a and b, the center of mass of the lamina described in problem 10 may or may not lie within the gnommon itself. For what values of a and b will the center of mass fall within the gnommon?