Math 52 - Autumn 2006 - Midterm Exam II
Problem 1. Let R be a region in R2 with area A, and coordinates of the centroid (a, b). Let
L denote the length of the boundary ∂R of the region R and (m, n) denote the coordinates
of the boundary’s centroid. Use (some or all of) the above symbols A, a, b, L, m and n to
find:
a)
Z
(x + y) ds
∂R
= L(m + n)
b)
I
(x + y) dy
∂R
ZZ
ZZ
∂x (x + y)dx dy =
=
R
dx dy = A
R
Problem 2. Calculate
I
z dx + x dy + y dz
C
where C is the intersection of the surfaces z = x2 and x2 + y 2 = 4 oriented clockwise as
viewed
R 2πfrom above.
Use 0 cos2 x dx = π.
Solution: Parametrize C as x(t) = 2 cos t, y(t) = −2 sin t and z(t) = x2 = 4 cos2 t. Then
I
Z 2π
z dx+xdy +y dz =
4 cos2 t·(−2) sin t+2 cos t·(−2) cos t+(−2) sin t·4·2 cos t sin tdt =
C
0
Z
=
2π
(−4) cos2 t dt = −4π
0
p
Problem 3. Let S be the part of the cone z = x2 + y 2 that lies between the planes z = 1
and z = 4.
a) Use the r and θ of cylindrical coordinates (r, θ, z) to parametrize S.
In cylindrical coordinates the equation of the cone is z = r. So:
ANSWER: x(r, θ) = r cos θ,
y(r, θ) = r sin θ,
z(r, θ) = r
For 1 ≤ r ≤ 4 and 0 ≤ θ ≤ 2π
b) Decide whether the perpendicular vector Nr,θ points up or down.
You may use a picture or algebra as your justification.
Tr points diagonally up and Tθ points horizontally in counterclockwise direction, so Tr × Tθ
points up or inside the cone.
1
c) For the above surface S oriented with normal vector pointing up (i.e. inside the cone),
use Stoke’s theorem to compute
ZZ →
∇ × F ◦ dS
S
→
for F = hyz, −xz, z 3y + ln(sin x2 )i.
By Stokes
ZZ
ZZ
→
∇ × F ◦ dS =
ZZ
→
∇ × F ◦ dS +
D−
1
S
→
∇ × F ◦ dS
D+
4
→
where D1− is the disk radius 1 centered at the origin and oriented with normal vector − k .
→
Similarly D4+ is the disk radius 4 centered at the origin and oriented with normal vector + k .
Now
→
→
→
i
→
j
k
→ →
∇ × F ◦ k = ∂x ∂y
◦ k = −z − z = −2z
∂z
3y
2
yz −xz z + ln(sin x ) Thus:
ZZ
→
∇ × F ◦ dS = −2(−1) · π(1)2 = 2π
D−
1
ZZ
→
∇ × F ◦ dS = (−2) · 4 · π(4)2 = −128π
D+
4
so the answer is −126π.
Indeed you do not need to differentiate ∂z (z 3y ) = z 3y ·
3y
.
z
Problem 4. For what values of the constants a and b will the vector field
→
2
2
2
F = hy + 2azx, bxy + azy, y + ax i
be conservative?
→
→
→
j
i
k
→
∇×F =
∂x
∂y
∂z
2
y + 2azx bxy + azy y 2 + ax2
= h2y − ay, −(2ax − 2ax), by − 2yi
→
which is 0 if a = 2 and b = 2.
Problem 5. Find the coordinates of the centroid of the first octant part of the sphere
x2 + y 2 + z 2 = 1, x ≥ 0, y ≥ 0, z ≥ 0.
Hint: By symmetry x̄ = ȳ = z̄.
2
The area (mass) of the surface is 18 · 4π. Parametrize S with spherical coordinates x =
sin φ · cos θ, y = sin φ · sin θ and z = cos φ with 0 ≤ φ ≤ π/2 and 0 ≤ θ ≤ π/2. Then:
Z
π/2
Z
π/2
Z
cos φ · sin φ dφ dθ = π/2 · 1/2 ·
z̄ · mass =
0
sin 2φ dφ = π/4
0
0
Thus x̄ = ȳ = z̄ = 1/2.
3
π/2