Mathematics 2210
EXAM III SOLUTIONS
Fall 2015
1. Evaluate the integral
ZZ
(x2 y)dA,
R
where R is the square with vertices (0, 0), (0, 1), (1, 0) and (1, 1).
Solution. Integral:
Z 1Z
1
(xy)dxdy
0 0
Answer:
1
6.
2. Evaluate
∞
Z
I=
2
e−x dx
−∞
Solution. Integral (after switch to polar):
Z 2πZ ∞
2
2
I =
e−r rdr
0
Answer:
√
0
π.
3. (a) Compute the surface area of the top part of the upside down
paraboloid
z = 4 − (x2 + y 2 ) that is cut off by the x-y plane.
Solution. Integral (Polar):
Z
2π
Z
2
r
Answer:
π
6 (17
√
0
p
4r2 + 1drdθ
0
17 − 1).
(b) Consider f (x, y) = (x − 2)2 + y 2 − 4. Find the global maximum
of f and where it occurs by using the gradient. Verify that it is
a maximum by using the second partial derivatives test. Be sure
to show all of your work.
Solution. ∇f =< 2x − 4, 2y >=< 0, 0 > which implies x =
2, y = 0 Then fxx = 2, fyy = 2 and fxy = 0. Thus fxx fyy − fxy =
4 > 0 and fxx = 2 > 0 so we have a minimum.
1
4. (a) Compute the Jacobian J(r, θ, z) of the transformation from cylindrical coordinates to Cartesian coordinates given below.
x = r cos θ
y = r sin θ
z=z
Solution.
The Jacobian

 matrix:
cos θ −r sin θ 0
 sin θ r cos θ 0 
0
0
1
Answer: J(r, θ, z) = r.
(b) Let ~v (x, y) = (y, 0). Find ∇ · ~v and ∇ × ~v . This vector field ~v
serves as a simplified model for what dangerous phenomenon?
Solution. ∇ · ~v = 0, ∇ × ~v = −k; the vector field serves as a
model for wind shear.
5. Use cylindrical coordinates to find the mass of the solid cylindrical shell
lying inside the cylinder r = b and outside the cylinder r = a (a < b)
from z=0 to z=1, if the density λ is proportional to the distance from
the origin, λ(r, θ, z) = kr where k ∈ R is a real constant.
Solution. Integral:
Z
0
2π
Z
1Z b
0
a
2
kr2 drdzdθ
Answer:
2π
3
3 k(b
− a3 ).
k
k
6. Let u(x, y, z) = p
= , where k > 0 is a real number.
2
2
2
r
x +y +z
(a) Find the gradient vector field associated with this potential by
computing F~ = −∇u. Express F~ in terms of ~r = (x, y, z). What
is the name of this famous law for the force field F~ ?
Solution. −∇u =
k~
r
||~
r||3
(b) What are the equipotential surfaces (level sets) of u(x, y, z)?
Solution. k/r = c ∈ R so r = k/c, constant radius spheres.
(c) Now let u(x, y, z) = 21 k(x2 + y 2 + z 2 ) with k > 0. Find the conservative force field F~ (x, y, z) = −∇u for any ~r = (x, y, z) ∈ R3 .
Express your result in terms of ~r. What is the name of this famous force law? Find the curl and divergence of F~ .
Solution. F~ = −∇u = −k(x, y, z) = −k~r. A conservative vector field always has curl = 0. Divergence of F~ is −k−k−k = −3k.
7. Evaluate the following line integral where C is determined by the parametric equation x = 2 cos t, y = 2 sin t, 0 ≤ t ≤ π/2.
Z
xy 2 ds.
C
Solution. Integral:
Z
π/2
p
8 cos t sin2 t 4 sin2 t + 4 cos2 tdt
0
Answer:
16
3 .
3