Name
ID
MATH 251
Exam 2 Version A
Sections 506
1-13
/52
Spring 2013
14
/12
P. Yasskin
15
/28
16
/12
Total
/104
Multiple Choice: (4 points each. No part credit.)
1. Compute
xy dA over the region R between the parabola y
x 2 and the line y
2x.
R
a. 4
b.
c.
d.
e.
3
8
3
32
15
10
3
29
6
2. Compute
3
0
9 x2
9 x2
x dy dx by converting to polar coordinates.
a. 36
b. 27
c. 18
d. 9
e. 3
1
3. Find the mass of the triangular plate with vertices
0, 0 ,
and
2, 6
2, 6
if the surface density
2
is
x .
a. 4
b. 6
c. 12
d. 16
e. 24
4. Find the center of mass of the triangular plate with vertices
a.
b.
c.
d.
e.
0, 0 ,
2, 6
and
2, 6
if the surface
x2.
density is
8 ,0
5
9 ,0
5
5 ,0
8
5 ,0
9
6 ,0
5
5. Which of the following integrals is NOT equivalent to
4
4 z
0
0
y
f x, y, z dx dy dz? The region is
0
shown.
a.
4
4 z
0
d.
e.
f x, y, z dy dx dz
x2
0
4
4 y
2
0
0
y2
b.
c.
4 z
4
y
f x, y, z dx dz dy
4 y
f x, y, z dz dx dy
0
0
0
2
4 x2
4 z
0
0
x2
2
4
4 y
0
x2
0
f x, y, z dy dz dx
f x, y, z dz dy dx
2
6. Find the area of one petal of the 8 petaled daisy r
a.
b.
c.
d.
e.
sin 4 .
2
4
8
16
32
7. Find the mass of the solid between the paraboloids z
density is
x2
y 2 and z
8
x2
y 2 if the volume
z.
a. 64
b. 32
c. 16
d. 8
e. 4
8. Compute
x2
y 2 z dV over the solid hemisphere 0
x2
y2
z2
2
a. 0
b. 8
3
c. 16
3
d. 64
9
e. 2
3
9. Which integral gives the arclength of the ellipse r
2
a.
3 2
2 sin 2 d
3 4
2 cos 2 d
3 2
2 cos 2 d
6 cos , 3 sin , 3 sin
?
0
2
b.
0
2
c.
0
2
2 3
d.
3 sin 2
d
0
2
54 d
e.
0
10. A helical thermocouple whose shape is the curve r t
in a pot of water where the temperature is T
the water as measured by the thermocouple.
fds
HINT: f ave
ds
a. 173
3 cos t, 3 sin t, 4t for 0 t 4 is placed
y 2 z °C. Find the average temperature of
16
c. 1000
160
d. 250
e. 50
x2
8
4
b. 50
41
2
40
8
4
11. Find a scalar potential, f x, y, z , for F x, y, z
Then compute f 2, 2, 2
2xy 2
2x
2xz, 2x 2 y
3z, x 2
3z 2
3y .
f 1, 1, 1 .
a. 0
b. 1
c. 7
d. 23
e. 25
12. If f
x2
y2
a.
f
0
b.
f
0
c.
d.
F
F
2z 2 and F
xz, yz, z 2 , which of the following is false?
0
0
e. None of the above. They are all true.
13. Compute
G dV for G
xz, yz, z 2
over the solid cylinder x 2
y2
25 with 0
z
4.
C
a. 800
b. 400
c. 200
d. 80
e. 40
5
Work Out: (Points indicated. Part credit possible. Show all work.)
12 points
14.
x 2 dA over the "diamond"
Compute
y
D
shaped region bounded by the curves
y
1
x3, y
3
x3, y
4
x3, y
HINT: Define curvilinear coordinates
y
u
x
3
and y
v
7
u, v
x3.
so that
3
x .
x
a.
2 pts
What are the boundaries in terms of u and v?
b.
3 pts
Find formulas for x and y in terms of u and v.
c.
4 pts
Find the Jacobian factor J
d.
1 pts
Express the integrand in terms of u and v.
e.
2 pts
Compute the integral.
x, y
u, v
.
6
15.
28 points
Consider the elliptical region, E, in the plane z
2
x y
4 oriented upwards.
2
x
y above the circle
2
HINT: This ellipse may be parametrized by R r,
a.
10 pts
Note:
r cos , r sin , 2
r cos
Find the normal vector N to the ellipse and its length
r sin
.
N .
N starts hard but simplifies!
b.
3 pts
Find the surface area of the ellipse.
c.
3 pts
Find the mass of the ellipse if the surface density is
d.
12 pts
If F
x2
yz, xz, z 2 , compute the surface integral
y2.
F dS
E
7
16.
12 points
Compute
G dS for G
xz, yz, z 2
over the cylinder x 2
y2
25 for 0
z
4
C
with outward normal.
8