Math 221: Chapter 14 Exam Study Guide

In-Class Exam

1. A plane region is described. Write the limits for the double integral for both orders of integration.
2. 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.
3. A surface is given.
1. 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.
2. Write a double integral to both rectangular and polar coordinates to find the area of the surface. Then evalute whichever one is easier.
3. 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.
4. 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.
5. Find the mass and center of gravity for the laminate bounded by the given equations having the indicated density.

Take Home Exam

1. 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

 # Pts 1a 1b 2a 2b 2c 2d 3a 3b 3c 4a 4b 4c 5 Total 4 4 6 6 6 6 8 6 6 6 4 4 12 78

Take Home Points

 # Pts 1a 1b 1c 1d 1e 1f Total 2 4 4 4 4 4 22