MATH 260

EXAM 3

1. (20 pt) a) Consider the change of variables:

for the double integral . Compute the Jacobian of this transformation and rewrite the double integral in polar coordinates.

b) As in part a), rewrite in terms of the cylindrical coordinates:

Where is a constant. (In other words, you must compute the Jacobian).

2. (20 pt) Find the moment of inertia of the solid bounded by the planes , and the cylinder about the -axis. (The solid has constant density K and ).

3. (20 pt) Find the volume of the solid enclosed by the sphere and the planes and with . When you are finished, check your answer by looking at a special case.

4. (20 pt) Find the surface area of a sphere of radius .

5. (20 pt) Evaluate the double integral where is the triangle with vertices (1,0), (0,2) and (0,0), by making the change of variables and .

6. (EXTRA CREDIT..10 pt) Find the moment of inertia of the ellipsoid about the z-axis (the ellipsoid has constant density K and volume ).