# Math 221: Chapter 15 Study Guide

1. Evaluate the iterated integrals. Two parts. Look at problems 15.1.1-12 and 15.2.1-10
2. Express the integral as an equivalent integral with the order of integration reversed. Do not evaluate. Look at problems 15.2.41-46
3. Evaluate a double integral over the region described. 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 the 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 15.5.23-24
8. Find the centroid of the region enclosed. Look at problems 15.6.5-10
9. Use the Theorem of Pappus to find the volume of the solid of revolution of the region about a line. Be sure you can find the distance between a point and a line. Look at problems 15.6.37-40
10. Use cylindrical coordinates to find the volume of a solid. Look at problems 15.7.5-8, 13-16, 31-36
11. Use spherical coordinates to find the mass of a solid. Look at problems 15.7.23-24, 31-36
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. Evaluate the integral by making an appropriate change of variables. Look at problems 15.8.31-34
14. Find a transformation that will transform the region in the xy-plane into the region in the uv-plane. Look at problems 15.8.27-30.
15. Write a double integral that can be used to find the area of the region. Write it as both a type I and type II integral. Do not evaluate the integral.

## Notes:

• You may place the conversion formulas between rectangular, cylindrical, and spherical coordinates onto notecards. You may not place examples onto the notecards.
• Although not explicitly stated with each problem, part of what I will be grading is the sketch of the region where appropriate.

## Points per problem

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