Math 208 (Lutz) - Quiz #6, 3/7/14
Name:
Directions: This quiz has 3 questions, for a total of 20 points. Solve each problem carefully. You must
show ALL of your work in order to earn full credit.
1. (6 points) Evaluate the following double integral using polar coordinates:
!
1
0
!
√
1−x2
√
− 1−x2
x dy dx.
2. (6 points) Set up, but do not evaluate, a triple integral in cylindrical coordinates that gives the
volume of the upper hemisphere of a sphere of radius 5.
3. (8 points) Use spherical coordinates
to find the volume of the region above the cone z =
!
and below the hemisphere z = 4 − x2 − y 2 . !
(Hint: In spherical coordinates, the cone z = x2 + y 2 is given by φ = π/4.)
!
x2 + y 2