Name:

Directions: Show your work! Answers without justification will likely result in few points. Your written work also allows me the option of giving you partial credit in the event of an incorrect final answer (but good reasoning). Indicate clearly your answer to each problem (e.g., put a box around it). Good luck!

Problem 1 (10 pts). For the function

1. Write all anti-derivatives of f.
2. Find the particular anti-derivative whose value at x=0 is 2.

Problem 2 (20 pts). Consider the solid obtained by rotating the region bounded by the function

the x-axis, and about the y axis. You are to set up the integral for the volume of the solid using two methods:

1. the method of cylindrical shells, and
2. the method of disks (washers - whatever!).

Use your calculator to compute the two integrals, and verify that they give the same answer. If they do not give the same answer, then indicate which you have the most confidence in!

Problem 3 (20 pts). We want to approximate the integral

using three different methods (with n=4 subintervals):

1. The left endpoint method
2. The right endpoint method
3. The midpoint method

Compare the approximations to the exact value for the integral, and comment on the results of your calculations.

Problem 4 (15 pts). For the following integrals, indicate the substitutions you would use to compute the integrals, and rewrite the integral using the substitution (if necessary - that is, if the integral is not obvious or undefined). No need to evaluate the integrals.

1. 

2. 

3. 

Problem 5 (15 pts).

1. Draw the region bounded by the two curves and , and
2. find the area of the region.

Problem 6 (10 pts). True/False:

1. The integral of a sum is the sum of the integrals.
2. If , then on [a,b].
3. If f is continuous on [a,b], then

   

4. Consider function f, continuous on interval [a,b]. The derivative of an

   anti-derivative of f on [a,b] is f.

5. The integral of a quotient is the quotient of the integrals.

Problem 7 (10 pts).

1. If the rate of increase of a bank account is given by r(t), what does the integral

   

   represent?

2. Suppose that rate can be written as

   

   What is the increase in the bank account over the 5 years from year 2 to year 7?