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 ).