# Math 221: Chapter 15 Study Guide

1. Sketch the region and evaluate the iterated double integrals. Two parts.
2. Rewrite the double integral over the region with the appropriate limits. You may use a type I or type II region, whichever you find easier. Do not evaluate the integral.
3. Evaluate the iterated triple integral.
4. Find the centroid of a region and the volume when the region is rotated about a line. The integrals you need are already evaluated, you need to know what to do with them.
5. Find a transformation that, when applied, will map the region R in the xy-plane into the region S in the uv-plane.
6. Use a double integral to find the volume of the solid.
7. Find the surface area. Find the equations of a tangent plane and the normal line to the surface at a given point.
8. Find the mass and center of gravity of the laminate.
9. Use cylindrical coordinates to find the volume of the solid.
10. Use spherical coordinates to find the volume of the solid.
11. Find the Jacobian.
12. Make an appropriate change of variables to evaluate the integral.
13. Find the velocity, speed, and acceleration of the vector valued function at the indicated point. Find the displacement and distance the particle traveled over the indicated interval.
14. Find the unit tangent, normal, and binormal vectors at the indicated point.

## Notes

• The first part of the exam, questions 1-5, must be worked without a computer and turned in before the second part of the exam can be taken.
• For part 2, questions 6-14, use Derive, Winplot, or DPGraph to assist. Sketch the region where it is needed to determine limits and write down the integral(s), but then use Derive to evaluate them.

## Points per problem

 # Pts 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Total 20 8 10 12 10 10 20 10 10 10 10 10 20 15 175