Math 212
Name: __________________________________
Homework 5
y
H0, 2L
1. Let W be the shaded region shown in the figure to the right.
(a) Find the equation for the horizontal line C œ " in polar
coordinates. Express your answer as an equation of the
form < œ 0 a)b.
H-1, 1L
H1, 1L
x
(b) Find a simple equation for the given circle in polar coordinates. Again, express your answer in
the form < œ 0 a)b.
(c) Use a double integral in polar coordinates to find the average value of < on the region W . Feel
free to use a computer or a table of integrals to help with the integration.
2. Let I be the solid defined by the following inequalities:
B#  C# Ÿ ",
B#  C#  D # Ÿ %,
Use cylindrical coordinates to evaluate ((( ˆB#  C# ‰D .Z .
I
and
D € !.
3. (a) Let B and C be two randomly chosen numbers in the interval c"ß "d. What is the probability
that C € B# ? (Hint: Think about B and C as the coordinates of a point in the plane.)
(b) Now suppose that B, C, and D , are three randomly chosen numbers in the interval c"ß "d. Use a
double or triple integral to compute the probability that D € B#  C# .
4. Let U be the quarter-sphere defined by the inequalities
B#  C#  D # Ÿ ",
C € !,
Use spherical coordinates to evaluate ((( CD .Z .
U
and
D€!