# Math 221: Chapter 14 Exam Study Guide

## In-Class Exam

- A plane region is described. Write the limits for the double integral for both orders of integration and then evaluate whichever one is easier.
- Transform the region R into a rectangular region S. Write the transformations and label the curves with their new equations. Find the Jacobian. Use the transformation to evaluate the integral.
- Find the mass and center of gravity for the laminate bounded by the given equations having the indicated density.
- Sketch the region or solid described. Write a double or triple integral using the indicated coordinate system that can be used to find the area or volume. Four parts: an area in polar coordinates and a volume in rectangular, cylindrical, and spherical coordinates. You will need to know the Jacobian for each coordinate system. Do not evaluate the integrals, only set them up.

## Take Home Exam

- Given a piecewise defined region, find the equations of each border, the area of the region, the centroid, the moments of inertia about the origin / pole, the moments of inertia about the centroid, and the center of pressure on a sail.
- Given a geometric figure, find the surface area, find the volume, find a point where a particular vector is normal to the surface, find the center of gravity.

## Notes

- There is a 30 point take home exam that is due the day of the in-class exam. You should use technology when completing the take home exam.
- You will need to know the Jacobians for transforming to polar, cylindrical, and spherical coordinate systems.
- This test has few problems worth a lot of points. Make sure you have mastered those topics.

## Points per problem

# |
1 |
2 |
3 |
4 |
TH1 |
TH2 |
Total |

Pts |
10 |
20 |
10 |
40 |
10 |
20 |
110 |