# Math 221: Chapter 14 Exam Study Guide

## In-Class Exam

1. Write the limits for the double integral for both orders of integration. Then integrate whichever one you find easier.
2. Find the surface area of the surface described.
3. Transform the region R into a rectangular region S. Write the transformations. Sketch the region S. Find the Jacobian. Use the transformation to evaluate the integral.
4. Find the indicated 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.

## Take Home Exam

1. Sketch the plane region for a solid and the solid. Then find the volume of the solid using double and triple integrals. Evaluate a triple integral over the solid.
2. Sketch the plane region for a solid and the solid. Then find the volume of the solid using double and triple integrals. Find the mass of the solid, the center of mass, and the moments of inertia.
3. 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 55 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

 # Pts 1a 1b 1c 2 3a 3b 3c 3d 4a 4b 4c 4d Total 3 3 3 6 3 1 3 3 5 5 5 5 45

### Take Home Points

 # Pts 1a 1b 1c 1d 1e 2a 2b 2c 2d 20 200 2000 3a 3b 3c 3d 3e 3f Total 2 2 3 3 3 2 2 3 3 3 3 6 2 2 4 6 4 4 55