Name
MATH 221
Exam 2
Sections 503
1-8
/48
Fall 2012
9
/12
P. Yasskin
10
/20
11
/20
Total
/100
Multiple Choice: (6 points each. No part credit.)
1. Compute
2
2
0
x
5y 3 dy dx.
a. 68
b. 48
c. 32
d. 27
e. 15
2. Which of the following is the polar plot of r
sin 3 ?
y
a.
b.
c.
1
0
2
4
6
x
-1
d.
e.
1
3. Find the mass of a triangular plate whose vertices are
0, 0 ,
1, 0
and
1, 3 , if the density is
2y.
a. 1
b. 2
c. 3
d. 4
e. 5
4. Find the y-component of the center of mass of a triangular plate whose vertices are
and
1, 3 , if the density is
0, 0 ,
1, 0
2y.
a. 9
b.
c.
d.
e.
5.
2
7
2
5
2
3
2
1
2
In spherical coordinates
centered at
2 cos
is a sphere of radius 1
0, 0, 1 . If its volume density is
x2
y2
z2
then its mass is given by the integral:
a. M
b. M
c. M
d. M
e. M
2
/2
2
0
2
0
0
2 cos
0
2
0
0
0
2
0
0
2 cos
0
2
0
0
0
0
/2
/2
2 cos sin d d d
4
2 cos
2 cos
sin d d d
2
2
sin d d d
sin d d d
4
sin d d d
0
2
6.
Find the area inside the circle r
and outside the cardioid r
a.
b.
1
3 cos
y
cos .
4
2
x
c.
3
2
e. 2
d.
7. Parabolic coordinates are given by u
So the area element is dA
a.
b.
c.
d.
e.
2 2
2 2
4 2
4 2
8 2
1
v
1
v
1
v
1
v
1
v
u
u
u
u
u
y
x 2 and v
y
x 2 where v
u.
dx dy
du dv
du dv
du dv
du dv
du dv
3
xe yz
8. If f
ye xz , then
f
2xe yz 2xe xz 2xyze yz , 0, 2ze xz 2ze yz
2xe yz 2xe xz 2xyze yz , 0, 2ze xz 2ze yz
e yz yze xz , xze yz e xz , xye yz xye xz
e yz yze xz , xze yz e xz , xye yz xye xz
e. 0
a.
b.
c.
d.
Work Out: (Points indicated. Part credit possible. Show all work.)
9. (12 points) Determine whether or not each of these limits exists. If it exists, find its value.
a.
b.
lim
3x 2 y 2
x 6 3y 3
lim
xy 2
x y2
x,y
x,y
0,0
0,0
2
4
10. (20 points) Compute
2
F dS for the vector field F
yz, xz, z 2
over the paraboloid
2
z 9 x y for z 5 oriented down and in.
Note: The paraboloid may be parametrized as R r,
r cos , r sin , 9
r2 .
5
11. (20 points) Compute
below the cone z
9
x3, y3, x2z
F dV for the vector field F
x2
y2
and above the plane z
y2z
over the solid region
5.
6