Math 52 - Winter 2006 - Final Exam
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.
There are ten problems with the total of 100 points. Point values are given in parentheses.
You have 3 hours (till 10PM) to answer all the questions.
Question
Score
Maximum
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
Total
100
Problem 1. (10 pts.) Evaluate
Z
0
1
Z
1
2
ex dx dy
y
1
Problem 2. (10 pts.) Let C be the ellipse
(x − 1)2 + (2y + 1)2 = 1
a) Compute
I
x dy
C
Use the formula:
Z
2π
cos2 x dx = π
0
b) Find the area bounded by the curve C.
2
Problem 3. (10 pts.) Show that for any 0 < a < b and 0 < c < d, the area of the region
bounded by the curves
x2 = ay,
x2 = by,
x3 = cy 2 ,
1
is 15
(b5 − a5 ) (c−3 − d−3 )
Hint: Perform the change of variables x2 = uy, x3 = vy 2 .
3
x3 = dy 2
Problem 4. (10 pts.) Let S be a part of the surface z 2 = 2xy bounded by the planes
x + y = 1, x = 0 and y = 0. Find
ZZ
√
xy ds
S
4
Problem 5. (10 pts.)
and (0, 1).
a) Find
Let T be the boundary of the triangle with vertexes (0, 0), (1, 0)
Z
(x + y) ds
T
b) Find
I
(x + y) dy
T
5
Problem 6. (10 pts.) Let C be a closed simple curve in R2 and D the region bounded by
C.
a) Show that
I
x2 dy = 2 · Area(D) · x̄
C
where x̄ is the x-coordinate of the centroid of D.
Hint: use Green’s theorem.
b) Find the corresponding formula expressing the y-coordinate of the centroid of D in terms
of area of D and a line integral over C.
6
ZZ
→
→
F ◦d s
Problem 7. (10 pts.) Use the divergence theorem to compute the outward flux
S
→
for F = h(y − x), (z − y), (z − x)i and S the cube bounded by the planes x = ±1, y = ±1
and z = ±1.
7
Problem 8. (10 pts.) Consider the integral
Z 1 Z 1 Z 1−y
dz dy dx
−1
x2
0
a) Sketch the region of integration.
b) Write the integral as an equivalent iterated integral in the order dy dz dx.
8
Problem 9. (10 pts.) Use Stoke’s theorem for evaluation of
ZZ →
∇ × F ◦ dS
S
p
→
for F = hyz, −xz, z 3 i and S being the part of the cone z = x2 + y 2 between the planes
z = 1 and z = 3, oriented with the normal vector pointing away from the z axis.
9
Problem 10. (10 pts.)
Out of the two integrals below evaluate the ONE to which
Fundamental Theorem of Line Integrals applies:
Z
(1,0)
a)
(0,−1)
x dy − y dx
(x − y)2
Z
(2,3)
b)
(3,2)
x dy − y dx
(x − y)2
You may use the formula (DO NOT PROVE IT!):
Z
x dx
−x
+ ln |x − a| + C
=
2
(x − a)
(x − a)
Indicate which integral you are computing and explain why you can not apply Fundamental
Theorem of Line Integrals to the other one.
10