Name
Sec
MATH 221
Exam 2
Section 500
Spring 2011
1-12
/60
14
/15
13
/15
15
/15
Total
/105
P. Yasskin
Multiple Choice: (5 points each. No part credit.)
1. Compute
a.
b.
c.
d.
e.
2
y
∫ 0 ∫ 0 xy dx dy.
1
2
3
4
y2
2. Find the area of one loop of the rose r = sin(3θ).
a.
b.
c.
d.
e.
π
12
π
6
π
4
π
3
π
2
3. Compute
∫∫∫ x 2 + y 2 dV
over the region between the cones z =
x2 + y2
and z = 4 − x 2 + y 2 .
a. 8π
b.
c.
d.
e.
3
16π
3
32π
3
16π
5
32π
5
1
4. Find the mass of the hemisphere x 2 + y 2 + z 2 ≤ 4 with y ≥ 0 if the density is δ = y.
a. π
b.
c.
d.
e.
2
π
2π
4π
8
5. Find the center of mass of the hemisphere x 2 + y 2 + z 2 ≤ 4 with y ≥ 0 if the density is δ = y.
a.
b.
c.
d.
e.
0, 64π , 0
15
0, 16 , 0
15
2
0, π , 0
12
0, 15 , 0
16
0, 122 , 0
π
6. Compute
∮ F⃗ ⋅ ds⃗
⃗ = (−16x 2 y, 9xy 2 ) counterclockwise around the ellipse
for F
HINTS: The ellipse may be parametrized by ⃗r(θ) = (3 cos θ, 4 sin θ).
Since sin(2θ) = 2 sin θ cos θ, we have 4 sin 2 θ cos 2 θ = sin 2 (2θ).
a.
b.
c.
d.
e.
2
x 2 + y = 1.
9
16
−864π
−288π
144π
288π
864π
2
3
3
π, π
is a critical point of the function f(x, y) = sin(x) cos(y) −
x+
y
3 6
4
4
Use the Second Derivative Test to classify this critical point.
7. The point
a. Local Maximum
b. Local Minimum
c. Inflection Point
d. Saddle Point
e. Test Fails
⃗ (x, y) =
8. Which of the following is the plot of the vector field F
a.
b.
d.
e.
1
x + y2
2
|x|, |y|
?
c.
3
⃗ (u, t) =
9. Find the area of the parameric surface R
ue t , ue −t , 2 u
for 0 ≤ u ≤ 2 and 0 ≤ t ≤ 1.
HINT: Look for a perfect square.
a. 2 2 e + 1
e −2
b. 2 2 e + 1
e
c. 2 2 e − 1
e −2
d. 2 2 e − 1
e
e. 2 e + 1
e
⃗ (u, t) =
10. Find the equation of the plane tangent to the parameric surface R
⃗ (2, 0) where u = 2 and t = 0.
point P = R
⃗ at u = 2 and t = 0.
Hint: Evaluate the normal N
ue t , ue −t , 2 u
at the
a. x + y − 2 z = −4 2
b. x + y − 2 z = 0
c. x + y − 2 z = 16 2
d. 2 x − 2 y + 2z = −4 2
e. 2 x − 2 y + 2z = 0
4
⃗ = (xy tan z, yz cos x, xz sin y), then ∇
⃗ =
⃗ ⋅∇
⃗ ×F
11. If F
a.
b.
c.
d.
e.
−2z cos y − 2x(tan 2 z + 1)
−2z cos y + 2x(tan 2 z + 1)
2z cos y − 2x(tan 2 z + 1)
2z cos y + 2x(tan 2 z + 1)
0
⃗ = (2xz − 3y, 8yz − 3x, x 2 + 4y 2 + 2z) for which f(0, 0, 0) = 0.
12. Let f be the scalar potential for F
Then f(1, 1, 1) =
a.
b.
c.
d.
e.
1
2
3
4
5
5
Work Out: (Points indicated. Part credit possible. Show all work.)
13. (15 points) The plane x + 2y + 4z = 8 intersects the 1st octant (x > 0, y > 0, z > 0) in a triangle.
Find the point on this triangle at which the function f = xy 2 z 3 is a maximum.
14. (15 points) Compute
z=
x2 + y2
∫∫∫ ∇⃗ ⋅ F⃗ dV
⃗ = (xy 2 , yz 2 , zx 2 ) over the solid above the cone
for F
below the sphere x 2 + y 2 + z 2 = 4.
6
15. (15 points) Compute
∫∫ E ∇⃗ × F⃗ ⋅ k̂ dx dy
⃗ = (−16x 2 y, 9xy 2 , 0) over the interior of the
for F
2
x 2 + y = 1.
9
16
⃗ ⋅ k̂ in rectangular coordinates.
⃗ ×F
HINTS: First compute ∇
Then compute the integral in elliptic coordinates x = 3u cos θ, y = 4u sin θ.
ellipse
7