MATH261 EXAM III SPRING 2015
NAME:
CSU ID:
SECTION NUMBER:
You may NOT use calculators or
any references. Show work to receive full credit.
GOOD LUCK !!!
Problem Points
1
18
2
18
3
18
4
26
5
20
Total
100
Score
In case they help, here are a few trig identities:
sin(2x) = 2 sin(x) cos(x),
cos(2x) = cos2 (x) − sin2 (x) = 2 cos2 (x) − 1 = 1 − 2 sin2 (x).
1. Consider the region in the positive orthant (where x > 0, y > 0, and z > 0) bounded
by z = 4 − y 2 , y = 2x, z = 0, and x = 0. Let f (x, y, z, ) = xyz.
(a) (6 pts) Set up, but do not evaluate, the integral of f (x, y, z) using dV = dzdydx.
(b) (6 pts) Redo (a) using dV = dydzdx.
(c) (6 pts) Redo (a) using dV = dxdzdy.
2. Consider the region R within the cylinder x2 + y 2 ≤ 4, bounded below by z = 0 and
above by z = 2 − y. Assume a mass density of δ = z.
(a) (10 pts) Set up and evaluate the integral representing the mass of the solid.
(b) (5 pts) Define, but do not evaluate, the integral representing Myz .
(c) (3 pts) Without evaluating any integrals, what is Mxz ? Explain clearly.
3. (18 pts) Using spherical coordinates set up, but do not evaluate, a triple integral
giving the volume of the region between the spheres x2 + y 2 + z 2 − 2z = 0 and
x2 + y 2 + z 2 − z = 0.
4. Consider
Z Z
(x + y)2 ex
2 −y 2
dA where R is the polygonal region with vertices
R
(1, 0), (0, 1), (−1, 0), (0, −1).
Let u = x + y, w = x − y. Complete the following:
(a) (8 pts) Find (x, y) in terms of (u, w).
(b) (8 pts) Draw the region R in the (x, y) plane and the corresponding region in the
(u, w) plane, with all corners clearly labeled with the coordinate point, e.g., (1,0).
(c) (5 pts) Compute the determinant of the Jacobian for this substitution.
(d) (5 pts) Set up the integral in the (u, w) variables, clearly showing the limits of
integration, but do not evaluate.
5. Consider the vector field F = hM, N, P i with
M = yz 2 cos(xy) + ln(z + 1),
N = xz 2 cos(xy) + 3 sin(z)e3y sin(z) ,
x
+ 3y cos(z)e3y sin(z) .
P = 2z sin(xy) +
z+1
(a) (9 pts) Without finding the potential function, show that F is conservative.
(b) (6 pts) Find the potential function, φ, for the force.
(c) (5 pts) Evaluate
R (0,π/2,π)
(π,1,0)
F · dr.