Each problem is worth 5 points. For full credit indicate clearly how you reached your answer.

1. Suppose that on some day the temperature at 3pm in the state of Kansas is 88 degrees at the southwest corner, 94 degrees at the southeast corner, and 90 degrees at the northeast corner. You may also assume that the temperature varies linearly, and that Kansas is close enough to being a rectangle 200 miles from south to north and 400 miles from west to east. Set up and use a double integral to compute the average temperature in Kansas.
 
 

2. Pat the mathematician runs a catering business during summer break. Pat is making a deli tray, and begins to ponder the volumes of irregular slices of sausage. Suppose the sausage is shaped like the cylinder x²+y²=1, with one cut made perpendicular to the cylinder along the plane z=0 and the other slice made along the plane z=mx+c (for some positive constants c and m, with c>m). Set up and evaluate an iterated integral for the volume of one of the wedges that results.
 
 

3. Alex the artist is a good friend of Pat's, and together they plan to create a mathematical sculpture involving several flat pieces of sheet metal shaped like the region bounded by the curves y=0, x=1, and y=xⁿ for various values of n which are at least 1. These metal pieces will all be hanging from the ceiling, so Alex and Pat need to know where to attach the supporting wires to make them balance. Compute the center of mass of such a shape (assuming constant density).
 
 

4. Alex and Pat also plan to create a Valentine's Day version of their hanging sculpture which involves a piece shaped like the cardioid r=1-sin . Find the center of mass of this piece.