Math 52 - Winter 2006 - Midterm Exam II
Name:
Student ID:
Signature:
Instructions: Print your name and student ID number, print your section number and
TA’s name, write your signature to indicate that you accept the honor code. During the
test, you may not use notes, books, calculators. Read each question carefully, and show all
your work.
There are —– problems with the total of ——— points. Point values are given in parentheses.
You have two hours (7PM-9PM) to answer all the questions.
Question
Score
Maximum
1
20
2
20
3
20
4
20
Total
80
Problem 1. Let S be the boundary of the solid cone z ≥ ha · r, z ≤ h. Find the coordinates
of the centroid of S.
You may use the formula
for the lateral surface area of the cone with hight h and radius a:
√
2
A = πal, where l = a + h2 .
Problem 2. Find the coordinates of the centroid of the boundary of the spherical triangle
x2 + y 2 + z 2 = R2 , x, y, z ≥ 0.
Problem 3. (10 pts.) Let for x > 0 C be a sum of two straight line segments: from the
point (1, 0) to the point (x, 0) and then from (x, 0) to (x, y).
a) Evaluate
Z
x dx + y dy
p
x2 + y 2
C
(your answer must depend on x and y).
b) Will the value of the integral
Z
C
x dx + y dy
p
x2 + y 2
change if C is replaced with any other curve contained in R2 \ (0, 0) strating at (1, 0) and
ending at (x, y)?
Either give an example or show that the value of the integral will not change.
Notice: In the above “any curve in R2 \ (0, 0)” means that this curve could enter the region
x ≤ 0 or even wind around the “bad” point (0, 0).
→
Problem 4. Let F = h−x2 y, xy 2 , sin x2 i and let S be part of the paraboloid z = 9 − x2 − y 2
above xy-plane with upper unit normal vector. Find
ZZ → →
∇ × F · n dS
S
Hint: use Stokes theorem.
1
Problem 5. (10 pts.) Let S be a surface cut from half–cylinder y 2 + z 2 = 9, z ≥ 0 by the
planes x = 0 and x = 4. Find the coordinates of the centroid of S.
Hint: Some of the coordinates can be found by the symmetry of S.
2
→
Problem 6. Let F = h2x · ln y − yz,
→
x2
y
− xz, −xyi.
a) Show that F is conservative vector field on some region of R3 . Describe that region.
b) Evaluate
Z
(2,1,1)
(2x · ln y − yz) dx +
(1,2,1)
3
x2
− xz
y
dy − xy dz
R →
→
Problem 7. Find ◦C F ·ds for F = hex sin y, ex cos yi and C the right-hand loop of the
graph of the polar equation r2 = 4 cos θ.
20
4