Name
ID
MATH 253
Exam 2
Sections 201-202
Fall 2006
P. Yasskin
Multiple Choice: (5 points each. No part credit.)
1. Compute
2
z
xz
2
4
∫0 ∫0 ∫0
1-9
/45
12
/12
10
/12
13
/12
11
/12
14
/12
Total
/105
15x dy dx dz.
a. 4
b. 8
c. 16
d. 32
e. 64
2. Compute
∫0 ∫y
2
y sin(x 2 ) dx dy by interchanging the order of integration.
a. − cos 16
b.
c.
d.
e.
2
cos 16 − 1
2
1 − cos 16
4
cos 16
8
cos 16 − 1
4
1
3. Which of the following is the polar plot of r = cos(3θ)?
a.
b.
y
c.
1
0
2
4
6
x
-1
d.
4.
e.
y
Find the area of the region inside the circle r = 4
outside the polar curve r = 3 + 2 cos(2θ) with y ≥ 0.
4
2
The area is given by the integral:
-4
5π/3
4
5π/3
3+2 cos(2θ)
5π/3
4
5π/6
4
5π/6
4
a. A =
∫ π/3 ∫ 3+2 cos(2θ) r dr dθ
b. A =
∫ π/3 ∫ 4
c. A =
∫ π/3 ∫ 3+2 cos(2θ) dr dθ
d. A =
∫ π/6 ∫ 3+2 cos(2θ) dr dθ
e. A =
∫ π/6 ∫ 3+2 cos(2θ) r dr dθ
-2
0
2
4
x
dr dθ
2
5.
Find the volume of the apple given
in spherical coordinates by ρ = 3ϕ.
The volume is given by the integral:
2π
a. 54π
∫0
b. 54π
∫ 0 ϕ 2 sin ϕ dϕ
c. 27π
∫0
d. 27π
∫ 0 ϕ 2 sin ϕ dϕ
e. 18π
∫ 0 ϕ 3 sin ϕ dϕ
ϕ 2 sin ϕ dϕ
π
2π
ϕ 2 sin ϕ dϕ
π
π
⃗ = (y − z, x + z, y − x + 2z).
6. Find a scalar potential f for the vector field F
Then evaluate f(1, 1, 1) − f(0, 0, 0):
a. 1
b. 2
c. 4
d. 5
e. 7
⃗ for any vector field F
⃗.
⃗ ×F
7. Which vector field cannot be written as ∇
⃗ = (xz, yz, z 2 )
a. A
⃗ = (x, y, −2z)
b. B
⃗ = (xz, yz, −z 2 )
c. C
⃗ = (z sin x, −yz cos x, y sin x)
d. D
⃗ = (x sin y, cos y, x cos y)
e. E
3
8. Find the total mass of a plate occupying the region between x = y 2 and x = 4
if the mass density is ρ = x.
a. 64
b.
c.
d.
e.
5
128
5
256
5
32
3
32
9
9. Find the center of mass of a plate occupying the region between x = y 2 and x = 4
if the mass density is ρ = x.
a.
b.
c.
d.
e.
20 , 0
7
12 , 0
5
512 , 0
7
128 , 0
5
14 , 0
5
4
Work Out: (12 points each. Part credit possible. Show all work.)
10.
Find the dimensions and volume of the largest box
which sits on the xy-plane and whose upper vertices
are on the elliptic paraboloid z = 12 − 2x 2 − 3y 2 .
You do not need to show it is a maximum.
You MUST eliminate the constraint.
Do not use Lagrange multipliers.
5
11. A pot of water is sitting on a stove. The pot is a cylinder of radius 3 inches and height 4 inches.
If the origin is located at the center of the bottom, then the temperature of the water is
T dV
T = 102 + x 2 + y 2 − z. Find the average temperature of the water:
T ave = ∫∫∫
.
∫∫∫ dV
6
12.
Compute
∫∫ x dx dy
over the "diamond"
D
y
6
shaped region bounded by the curves
y = x2
y = 3 + x2
y = 4 − x2
y = 6 − x2
Use the curvilinear coordinates
u = y + x 2 and v = y − x 2 .
(Half credit for using rectangular coordinates.)
4
2
0
0.0
0.5
1.0
1.5
2.0
x
7
13. The sides of a cylinder C of radius 3 and height 4 may be parametrized by
R(h, θ) = (3 cos θ, 3 sin θ, h) for 0 ≤ θ ≤ 2π and 0 ≤ h ≤ 4.
⃗ for F
⃗ ⋅ dS
⃗ = (−yz 2 , xz 2 , z 3 ) and outward normal.
⃗ ×F
Compute ∫∫ ∇
C
⃗ = ⃗e h × ⃗e θ , ∇
⃗ and
⃗ ×F
HINT: Find ⃗e h , ⃗e θ , N
⃗
⃗ ×F
∇
⃗ (h, θ) .
R
8
14. The hemispherical surface x 2 + y 2 + z 2 = 9 has surface density ρ = x 2 + y 2 .
⃗ (ϕ, θ) = (3 sin ϕ cos θ, 3 sin ϕ sin θ, 3 cos ϕ).
The surface may be parametrized by R
Find the mass and center of mass of the surface.
⃗ = ⃗e ϕ × ⃗e θ ,
HINT: Find ⃗e ϕ , ⃗e θ , N
⃗
N
⃗ (ϕ, θ) .
and ρ R
9