# Math 221: Chapter 15 Study Guide

1. Evaluate the iterated integrals. Two parts. Look at problems 15.1.1-12 and 15.1.1-10.
2. Sketch the region and then express the integral as an equivalent integral with the order of integration reversed. Do not evaluate. Look at problems 15.2.41-46.
3. Use a double integral to find the volume of the solid. Look at problems 15.2.31-38.
4. Evaluate the iterated integral by converting to polar coordinates. Look at problems 15.3.23-30.
5. Find the surface area. Look at problems 15.4.35-46.
6. Find an equation of the tangent plane to the parametric surface at the stated point. Look at problems 15.4.29-34.
7. Sketch the solid whose volume is given by the integral and then find the volume. Look at problems 15.5.1-8.
8. Find the centroid of the region. Look at problems 15.6.5-10.
9. Use cylindrical coordinates to find the volume of the solid. Look at problems 15.7.5-8.
10. Use spherical coordinates to find the volume of the solid. Look at problems 15.7.9-12, 31-36.
11. Use spherical or cylindrical coordinates to evaluate the integral. Look at problems 15.7.13-16.
12. Solve for x and y in terms of u and v and then find the Jacobian. Look at problems 15.8.5-8.
13. Make an appropriate change of variables to evaluate the integral. Look at problems 15.8.31-34.
14. Find a transformation to change a region R in the xy plane into a region S in the uv plane. Look at problems 15.8.27-30.
15. Write both type I (dy dx) and type II (dx dy)double integrals that can be used to find the area of the region R. Do not evaluate. Look at probems 15.2.13-16.
16. Use the Theorem of Pappus to find the volume of the solid of revolution. Look at problems 15.7.40-41.

## Notes

• Some of the problems may be directly from the text.
• For all problems involving integrals except #1, you should write the integral on your test and then use Derive to evaluate it. You will not have time to finish the test if you try to evaluate all the integrals by hand.
• You should make a sketch of the region whenever you are creating an integral from a description of a region, surface, or solid.

## Points per problem

 # Pts 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Total 10 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 100