Name
MATH 253
Exam 2
Sections 201,202
Fall 2014
Solutions
P. Yasskin
Multiple Choice: (5 points each. No part credit.)
2
1. Compute
0
a. 1
b.
c.
d.
e.
6
1
12
1
20
1
6
1
12
0
d d d .
0
5
5
Correct Choice
5
4
4
2
Solution:
0
0
3
d d d
0
0
3
d d
3
0
2. Which of the following is the polar plot of r
3
2
y
y
-5
-4
-3
-2
-1
1
3
y4
c.
2
x-1
12
0
5
4
b.
12
5
?
3 cos
2
a.
3
0
0
4
d
2
1
-4
-2
-3
-2
-1
1
2
-2
2
4
x
3
x
-3
Correct Choice
2
y
-2
-1
0
1
x
2
y
1
-1
d.
0
-4
-3
-2
-1
e.
-2
x
-1
-3
-2
-4
Solution: c is the rectangular plot not the polar plot. Notice when
loop at x
1.
0, r
1, which starts the inner
1
3. A plate fills the region between the graphs of x
|y| and x
4. Find its mass if its surface density
2
is
x .
a. 256
3
128
3
256
128
64
b.
c.
d.
e.
Correct Choice
4
Solution: M
x
0
4
x 2 dy dx
dA
x
0
4. A plate fills the region between the graphs of x
c.
d.
e.
4
Solution: M y
My
M
x
5.
dx
x3
0
|y| and x
a.
b.
c.
d.
e.
x
x x 2 dy dx
x dA
0
2048
5
1
128
x
4
x3y
0
4
x
y
x
dx
x4
x 4 dx
0
5
2x
5
4
0
2048
5
16
5
y
2 cos 3 .
12
x
6
Correct Choice
3
2
3
4
3
Solution: Find where the radius is zero: 2 cos 3
/6
A
128
0
4. Find the x-component of its
Find the area of one leaf of the rose
r
4
4
2x
4
x 3 dx
Correct Choice
5
32
5
512
5
2048
5
5
2048
b.
x
x2.
center of mass if its surface density is
a. 16
4
x
y
x2y
2 cos 3
1 dA
/6
r dr d
/6
/6
2
/6
0
cos 2 3 d
/6
/6
2
/6
1
2
r
2
cos 6 d
2
0
2 cos 3
d
r 0
1
2
sin 6
6
cos 3
0
/6
2 cos 3
3
2
2
6
d
/6
/6
/6
3
2
6. Set up, but do not compute, the integral
F dV over the solid region between the paraboloid
V
2x 2
z
2y 2 and the plane z
2
8
0
2r 2
2 8
a.
0
2
b.
0
2
2r 2
8
0
2
0
8
2r 2
z/2
0
2
0
8
0
z/2
0
0
0
d.
e.
y2 .
4r 2 dz dr d
0
2
c.
x3, y3, z x2
8 where F
4r 2 dz dr d
4r 3 dz dr d
Correct Choice
4r 2 dr dz d
4r 3 dr dz d
Solution: Take the divergence in rectangluar coordinates, then use cylindrical coordinates.
3x 2
F
3y 2
x2
y2
2
2
8
0
0
2r 2
F dV
V
4 x2
y2
4r 2
4r 2 r dz dr d
7. Find the average value of f
z on the solid hemisphere 0
1
Note: The average value of a function on a solid is f ave
V
a. 3
8
b.
z
x2
9
V
y2 .
f dV.
2
c. 3
2
d. 1
2
e. 9
Correct Choice
8
1 4 r3
2 33
Solution: V
2 3
3
2
V
f ave
/2
3
f dV
0
1
V
2
cos
V
0
18 . In spherical coordinates, f
sin d d d
sin 2
2
2
0
f dV
34
4
1
18
/2
0
z
3
4
4
cos .
34
4
0
9
8
8. Compute the line integral
F ds counterclockwise around one loop of the helix
r
4 cos , 4 sin , 3
for the vector field F
xz, yz, z 2 .
Hint: Compute
F in rectangular coordinates before integrating.
a.
b.
c.
d.
e.
16
32
64
128
0
Correct Choice
î
Solution:
F
x
k
y
xz yz z
î0
z
y
0
x
2
v
4 sin , 4 cos , 3
k0
y, x, 0
4 sin , 4 cos , 0
2
F ds
2
F vd
0
0
16 sin 2
16 cos 2 d
2
16 d
32
0
3
Work Out: (Points indicated. Part credit possible. Show all work.)
9. (20 points) A rectangular solid box is sitting on the xy-plane with its upper 4 vertices on the
ellipsoid x 2 4y 2 9z 2 108. Find the dimensions and volume of the largest such box.
Full credit for solving by Lagrange multipliers.
Half credit for solving by Eliminating a Variable.
50% extra credit for solving both ways.
Solution: Maximize: V
Note: x
4xyz.
0 y
0 z
V
g
0 so the volume will not be zero.
Method 1: Lagrange Multipliers:
The constraint is g
V
4yz, 4xz, 4xy
2x
2yz
x
4yz
g
x2
4y 2
L
x2
x2
12,
g
4xz
8y
2xy
xz
9z
2y
x 2 108
y
3
9z 2
W
6,
H
36
4y 2
9z 2
108
2x, 8y, 18z
4xy
18z
4y 2
x2
x2
9z 2
36 x
36
z
2,
x2
6
2
V
4xyz
LWH
4 6 3 2
144
Method 2: Eliminate a Variable:
V
4xyz
4yz 108
Vy
4z 108
Vz
4y 108
4y 2
4y 2
4y 2
y 4y
108
4y 2
9z 2
z 9z
12y
108
12,
2
0
y
3
4y 2
9z 2
108
W
6,
H
0
9z 2
4yz 18z
9z 2
9z 2
L
4y 2
2 108
4y 2
x2
4yz 8y
9z 2
108
108
9z 2
4y 2 9z 2
108 8y 2
2 108
0
0
9z
2
36
2,
4y 2
108
108
8y
36
36
V
4xyz
2
0
9z 2
18z 2
108
x
0
72
216
108
4y 2
4z
9z 2
108
4y 2
4y
9z 2
16y 2
18z 2
0
0
36
z
2
6
LWH
4 6 3 2
144
4
10.
(20 points) Compute the integral
y dA over
y
the region in the first quadrant bounded by
1 , and y
1 .
y x 2 , y 8x 2 , y
x
8x
Use the following steps:
2
1
0
0.0
u 3 x 2 and y
a. Define the curvilinear coordinates u and v by y
0.5
1.0
x
v3 .
x
Express the coordinate system as a position vector.
v3
v
x
x
u
x u, v , y u, v
u3x2
r u, v
u3x2
y
v
2
u , uv
2
u3 v 2
u
uv 2
b. Find the coordinate tangent vectors:
eu
r
u
v , v2
u2
ev
r
v
1 , 2uv
u
c. Compute the Jacobian factor:
x, y
u, v
J
2v 2
u
v2
u
3v 2
u
d. Compute the integral:
1
y dA
1/2
2
2
uv 2 3v
u du dv
1
2
1
1 du
1
1/2
3v 4 dv
u
2
1
3v 5
5
1
1/2
3 1
5
1
32
93
160
5
11. (20 points) Compute the flux
F dS of the vector field
H
x
F
x2
2
x
z2
y2
y
z
outward through the upper half of the sphere
y
x y
x y2
9. Use the following steps:
2
,
2
2
,
2
a. Parametrize the hemisphere:
R ,
3 sin cos ,
3 sin sin ,
3 cos
b. Find the tangent vectors:
e
3 sin sin ,
3 sin cos ,
0
e
3 cos cos ,
3 cos sin ,
3 sin
c. Find the normal vector:
î 9 sin 2 cos
9 sin 2 sin
k 9 sin cos sin 2
9 sin 2 cos , 9 sin 2 sin , 9 sin cos
N
9 sin cos sin 2
d. Fix the orientation of the normal (if necessary):
9 sin 2 cos ,
N
9 sin 2 sin ,
9 sin cos
e. Evaluate the vector field on the cylinder:
x2
y2
9 sin 2 cos 2
9 sin 2 sin 2
3 sin cos
,
3 sin
F R ,
3 sin sin
,
3 sin
3 sin
3 cos
3 sin
cos ,
sin ,
cos
sin
f. Evaluate the dot product:
F N
9 sin 2 cos 2
9 sin 2 sin 2
9 cos 2
9 sin 2
9 cos 2
9
g. Calculate the flux:
/2
F dS
H
2
F Nd d
9d d
0
0
9
2
2
9
2
6