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

1. Suppose that in some year the average wheat harvest in the state of Kansas is 3000 bushels per square mile at the southwest corner, 8000 bushels per square mile at the southeast corner, and 10000 bushels per square mile at the northeast corner. You may also assume that the wheat production 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 total wheat production of Kansas.
 
 

2. If [x] denotes the greatest integer less than or equal to x, evaluate the integral  where R = {(x,y)|1  x  3, 2  y  5}.
 
 

3. Suppose we take the cylinder x²+y²=r² and slice out a small wedge by cutting it along the planes z=0 and z=mx (for some constant m). Set up and evaluate an iterated integral for the volume one one of the wedges that results.
 
 

4. Set up an iterated integral and evaluate it to find the volume of the solid bounded by the surfaces z=4-x², z=x²-4, y=4-x², and y=x²-4.