MATH261 EXAM III FALL 2014
NAME:
SI:
SECTION NUMBER:
You may NOT use calculators or
any references. Show work to receive full credit.
GOOD LUCK !!!
Problem Points
1
15
2
15
3
25
4
23
5
22
Total
100
Score
1. Consider the solid bounded below by the plane z = 0, above by z = e−(x
the sides by x2 + y 2 = e−2 .
2 +y 2 )/8
and on
(a) (5 pts) Graph the volume.
(b) (10 pts) Using a double integral, find the volume of the solid.
2. Consider the volume in the first octant bounded by the planes 3x + 4y + 16z = 12 and
3x + y = 3.
(a) (5 pts) Graph the volume.
(b) (10 pts) Set up the integral(s) for the volume when dV = dz dx dy. Do Not
Evaluate.
3. Consider the mass with volume
below by
p
√ bounded above by the plane z = 2, bounded
2
the sphere ρ = 1 with z ≥ 2/2, and bounded on the sides by z = x + y 2 with
density δ = 1/(x2 + y 2 + z 2 ).
(a) (5 pts) Graph the region in the yz plane.
(b) (10 pts) Set up the integral for the mass in spherical coordinates.
(c) (5 pts) Evaluate the integral.
(d) (5 pts) What are x̄ and ȳ? Give a reason.
u + 32 w. Verify that
4. (a) (5 pts) Consider the mapping x = 25 u + 12 w + 1, y = −1
2
u = 38 x − 18 y − 38 and w = 81 x + 58 y − 18 . Next, consider the triangle in the x, y
plane with vertices (4, 1), (3, −2), and (1, 0). What are the vertices in the u, w
plane?
(b) (6 pts) Graph the triangular area R in the (x, y) plane and R0 in the (u, w) plane
described above. The graphs must clearly show the vertices defined in the form
(a, b).
(c) (6 pts) Find the Jacobian for the mapping R to R0 .
ZZ
(d) (6 pts) Using the information from above, set up
(2x+10y −2)2 (3x−y −3) dA
in the new coordinate system. Do Not Evaluate.
2
R
2
5. (a) (12 pts) Show that F = hz 2 exz + y, x − z sin(yz), 2xzexz − y sin(yz)i is a conservative force without computing the potential function.
R
(b) (10 pts) Evaluate C F · dr where C is a path from the point (0, π, 2) to the point
(π, 3, 0).