MATH 261 EXAM III FALL 2015
NAME:
CSU ID:
SECTION NUMBER:
Problem Points
1
18
2
24
You may NOT use calculators or
any references. Show work to receive full credit.
3
18
4
18
GOOD LUCK !!!
5
22
Total
100
Potentially useful conversions:
x = r cos(θ)
y = r sin(θ)
x = ρ sin(φ) cos(θ)
y = ρ sin(φ) sin(θ)
z = ρ cos(φ)
r = ρ sin(φ)
Score
1. (18 pts; 9 pts per part) Let R be the tetrahedron in the first octant bounded by the
coordinate planes and the plane passing through (1, 0, 0), (0, 1, 0), and (0, 0, 2) with
equation 2x + 2y + z = 2, as shown below. Using rectangular coordinates, set up the
triple integral to find the volume of R in each of the two following variable orders, but
DO NOT EVALUATE.
(a)
Z Z Z
1 dxdydz
(b)
Z Z Z
1 dzdydx
z
y
x
2. (24 pts; 12 pts per part) Consider the region S inside the cylinder x2 + y 2 = 1, outside
the sphere x2 + y 2 + z 2 = 1, and with z between 0 and 1. A view from above and a
cross section (looking along the positive x axis) are shown below.
z
x
z
y
y
(a) Set up one or more triple integrals to integrate a function f (x, y, z) over S using
cylindrical coordinates (using order dz dr dθ), but DO NOT EVALUATE.
(b) Set up one or more triple integrals to integrate a function f (x, y, z) over S using
spherical coordinates (using order dρ dφ dθ), but DO NOT EVALUATE.
3. (18 pts, as indicated) Consider the thin plate T consisting of the portion ofpthe unit
disk in the first quadrant of the (x,y)-plane (as shown below). Let δ(x, y) = x2 + y 2
be the density of T .
y
x
(a) (12 pts) Set up but DO NOT EVALUATE an integral to find the mass of T
using polar coordinates.
3
(b) (6 pts) For this problem, the x-coordinate of the center of mass is
. What is
2π
the y-coordinate of the center of mass? Be sure to provide an explanation
or a computation in support of your answer.
4. (18 pts, as indicated) The two parts of this problem are both about substitution but
do not depend on each other.
(a) (10 pts) Suppose we want to replace x and y with u and v using the substitution
(
x = u2 + v 2 ,
y = 3uv − 2v 2 .
Compute J, the determinant of the Jacobian for this substitution.
(b) (8 pts) Suppose we want to integrate f (x, y) = x + y over the diamond U with
boundary lines y = x + 1, y = x − 1, y = −x + 1, and y = −x − 1. Fill in the
three blank spaces below (two bounds and the integrand) to write the integral of
f (x, y) = x + y as an integral in u and v, using substitution x = u + v + 1,
y = u − v + 2. The first and third bounding lines above have already been
converted into (u, v) bounds and included below. DO NOT EVALUATE this
integral. (Hint: You may use the fact that J = −2 for this substitution.)
Z
Z
−1
dudv
0
5. (22 pts, as indicated) This problem is all about vector fields. The three parts are
independent.
(a) (6 pts) Determine (without searching for a potential function) whether the following vector field is conservative. Your answer should be yes or no, followed by
your (brief) reasoning.
F = hx2 − yx, y 2 − xy, z 2 − xyi
(b) (8 pts) G = h2xy 3 z 4 , 3x2 y 2 z 4 , 4x2 y 3 z 3 + 2i is a conservative vector field. Find the
potential function g(x, y, z) such that g(0, 0, 0) = 0.
(c) (8 pts) h(x, y, z) = x2 + y 2 + z 2 is a potential function for conservative vector field
H = h2x, 2y, 2zi. C is an unknown curve joining (0, 0, 0) to (1, 2, 3). Compute
the work done in moving along C in the presence of H.