Math 230 Calculus II Summer 2004

FINAL EXAM

13 problems, 10 points each

Provide complete explanations of all solutions

1. Evaluate the integral

2. Evaluate the integral:

3. Evaluate

4. Let (a) Use the graph of to estimate (b) Estimate and compare it to .

5. The region bounded by the graphs of and is revolved about the -axis. Find the surface area of the solid generated.

6. Find (by integration) the length of a circle of radius

7. Find a parametric or polar representation of a curve similar to the one below, a spiral wrapping around a circle - from the inside! (no proof necessary):

8. (a) State the definition of the sum of a series. (b) Use (a) to show that the series converges.

9. Test the following series for convergence (including absolute/conditional):

10. Find the radius and the interval of convergence of the series

11. Find the power series representation of the function and its interval of convergence.

12. Find the Taylor polynomial of order centered at of the function

13. Indicate which the following statements below is true or false (no proof necessary):

1. If the series and diverges then the series also diverges.

2. The series converges.

3. If and converges, then also converges.

4. The series converges.

5. The sequence converges to

Bonus Problem (10 points). Test for convergence:

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