MATH 221 Name________________

Instructions: Answer all the questions on the paper provided and show all your work so that partial credit may be given. You may use a calculator on any problem.

1. Find f(x) given that f(4) = 4 and . (15)

2. Let R be the region bounded by the graphs of . For each of the following, find the indicated quantity and write down the integral (or integrals) that is needed to find the quantity. (30)

a) The Area of region R.

b) The Volume that results when R is rotated about the x-axis.

c) The Volume that results when R is rotated about the y-axis.

3. A rectangular box with a top is to have a base that is two times as wide as it is long. If the box is to have a total surface area of 300 square inches what is its maximum volume? (20)

4. Using the substitution transform the integral into an integral in the variable u. Do not evaluate the integral. (10)

5. Evaluate the integral . (10)

6. Show that without evaluating the integral. (10)

7. If G(x) = , what is (5)