# 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 and then evaluate whichever one is easier.
2. 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.
3. Find the mass and center of gravity for the laminate bounded by the given equations having the indicated density.
4. 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

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.
2. 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

 # Pts 1 2 3 4 TH1 TH2 Total 10 20 10 40 10 20 110