Name
MATH 221
Exam 2
Section 503
Fall 2009
P. Yasskin
1-13
/65
15
/10
14
/15
16
/15
Total
/105
Multiple Choice: (5 points each. No part credit.)
1. Compute
2
x
9
3
∫ 0 ∫ −x 3y 2 dy dx.
a. 8
b. 12
c. 16
d. 24
e. 0
2. Compute
∫0 ∫
y
π sin(πx 3 ) dx dy.
HINT: Reverse the order of integration.
a. 1
b.
c.
d.
e.
9
2π
9
1
3
2
3
2π
3
1
3.
Find the area inside the circle r = 2 cos θ
but outside the circle r = 1.
a. π − cos π
b.
c.
d.
e.
3
3
2π + cos 2π
3
3
π + sin 2π
3
3
2π − sin π
3
3
2π + sin π
3
3
4. Compute
2
∫0 ∫0
4−x 2
ex
2 +y 2
dy dx.
HINT: Switch to polar coordinates.
a. π e 2
b.
c.
d.
e.
2
π e4
2
π (e 2 − 1)
2
π (e 4 − 1)
2
π (e 4 − 1)
4
2
5. Compute the mass of the solid cone
x 2 + y 2 ≤ z ≤ 4 if the volume density is ρ = z.
a. 4π
b. 8π
c. 16π
d. 32π
e. 64π
6. Compute the center of mass of the solid cone
a.
b.
c.
d.
e.
x 2 + y 2 ≤ z ≤ 4 if the volume density is ρ = z.
0, 0, 5
16
0, 0, 16
5
0, 0, 1024π
5
0, 0, 5
1024π
0, 0, 8
5
7. Find the average value of the function f =
1
x2 + y2 + z2
spheres x 2 + y 2 + z 2 = 1 and x 2 + y 2 + z 2 = 4.
over the solid region between the two
f dV
HINT: f ave = ∫∫∫
∫∫∫ 1 dV
a. 3
4
b. 5
8
c. 3
7
d. 4π
e. 8π
3
2
2
y
∫∫ cos x9 + 4 dx dy over the region inside the ellipse
HINT: Elliptical coordinates are x = 3t cos θ, y = 2t sin θ.
8. Compute
2
x 2 + y = 1.
9
4
a. 6π sin(1) − 6π
b. 6π sin(1)
c. 6π − 6π cos(1)
d. −6π cos(1)
e. 6π cos(1)
9. Compute
2 ,2
∫ (0,0)
x ds along the parabola y = x 2 parametrized as ⃗r(t) = (t, t 2 ).
a. 13
b.
c.
d.
e.
6
9
4
27
4
1 (17 3/2 − 1)
12
17 3/2
12
10. Compute
2 ,2
∫ (0,0)
⃗ f ⋅ ds⃗ for f = xy along the parabola y = x 2 parametrized as ⃗r(t) = (t, t 2 ).
∇
a. 2
b. 2 2
c. 4
d. 4 2
e. 8
4
⃗ = (−yz 2 , xz 2 , z 3 ) counterclockwise around the circle x 2 + y 2 = 4 with
for F
HINT: Parametrize the circle.
11. Compute
z = 4.
∫ F⃗ ⋅ ds⃗
a. 64
b. 128
c. 64π
d. 128π
e. 256π
12. Compute
∫∫∫ ∇⃗ ⋅ F⃗ dV
⃗ = (xy 2 , yz 2 , zx 2 ) over the solid hemisphere x 2 + y 2 + z 2 ≤ 25 with z ≥ 0.
for F
a. 125 π
3
b. 250 π
3
c. 500 π
3
d. 1250π
e. 2500π
13.
y
Which of the following vector fields
are plotted at the right?
a. (y 2 , 4x)
4
2
-4
-2
2
b. (−y 2 , 4x)
-2
c. (4x, y 2 )
-4
4
x
d. (4x, −y 2 )
e. (4xz, 4y)
5
Work Out: (Points indicated. Part credit possible. Show all work.)
14. (15 points) Find the volume of the largest box such the length plus twice the width plus three times
the height is 36.
⃗ (p, q) = (p, p + q 2 , p − q 2 ) for 0 ≤ p ≤ 3 and 0 ≤ q ≤ 2.
15. (10 points) Find the area of the surface R
⃗e p =
⃗e q =
⃗ =
N
⃗ =
N
A=
6
∫∫ P ∇⃗ × F⃗ ⋅ dS⃗
⃗ = (−yz 2 , xz 2 , z 3 ) over the paraboloid z = x 2 + y 2 with
for F
⃗ (r, θ) = (r cos θ, r sin θ, r 2 ).
z ≤ 4, oriented up and in, and parametrized by R
16. (15 points) Compute
⃗ =
⃗ ×F
∇
⃗
⃗ ×F
∇
⃗ (r, θ)
R
=
⃗e r =
⃗e θ =
⃗ =
N
∫∫ P ∇⃗ × F⃗ ⋅ dS⃗ =
7