# Math 221: Chapter 15 Study Guide

- Sketch the region and evaluate the iterated double integrals. Two parts.
- 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.
- Evaluate the iterated triple integral.
- 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.
- Find a transformation that, when applied, will map the region R in the xy-plane into the region S in the uv-plane.
- Use a double integral to find the volume of the solid.
- Find the surface area. Find the equations of a tangent plane and the normal line to the surface at a given point.
- Find the mass and center of gravity of the laminate.
- Use cylindrical coordinates to find the volume of the solid.
- Use spherical coordinates to find the volume of the solid.
- Find the Jacobian.
- Make an appropriate change of variables to evaluate the integral.
- 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.
- 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

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

Pts |
20 |
8 |
10 |
12 |
10 |
10 |
20 |
10 |
10 |
10 |
10 |
10 |
20 |
15 |
175 |