# 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.
- Write a double or triple integral that can be used to find the area or volume. Use the coordinate system indicated. 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.
- A surface is given.
- Write a double integral in both rectangular and polar coordinates to find the area of the region under the surface. Then evalute whichever one is easier.
- Write a double integral to both rectangular and polar coordinates to find the area of the surface. Then evalute whichever one is easier.
- Write a double and triple integral to find the volume of solid bounded above by the surface. Switch to either polar or cylindrical coordinates before finding the volume.

- 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.

## 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.

## Notes

- There is a 22 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.

## Points per problem

### In-Class Points

# |
1a |
1b |
2a |
2b |
2c |
2d |
3a |
3b |
3c |
4a |
4b |
4c |
5 |
Total |

Pts |
4 |
4 |
6 |
6 |
6 |
6 |
8 |
6 |
6 |
6 |
4 |
4 |
12 |
78 |

### Take Home Points

# |
1a |
1b |
1c |
1d |
1e |
1f |
Total |

Pts |
2 |
4 |
4 |
4 |
4 |
4 |
22 |