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 ).