Math 208 (Lutz)
Exam III Practice Problems
12/04/13
1. Calculate the flux of F! = xy!i + yz!j + zx!k out of the closed box 0 ≤ x ≤ 1, 0 ≤ y ≤ 1,
0 ≤ z ≤ 1 in two ways: directly and using the Divergence Theorem.
2. Use Green’s Theorem to evaluate the circulation of F! = 3y!i + xy!j around the unit circle (in
2-space), oriented counterclockwise.
3. Let S be the cylinder of radius 1 centered on the z axis and between the planes z = 0 and
z = 6, oriented outward. Calculate the flux of F! = xz!i + yz!j + z 3!k through S.
! = xy!i + z!j + 3y!k around the square of
4. Use Stokes’ Theorem to valuate the circulation of G
side 6, centered at the origin, lying in the yz-plane, and oriented counterclockwise viewed
from the positive x-axis.
5. Find a parameterization of the surface obtained by revolving the graph of z = (x + 1)−1/2
about the x-axis, for 0 ≤ x ≤ 8.
6. Calculate the line integral of F! = y(x + 1)−1!i + ln(x + 1)!j along C, where C is the curve
from the origin along the x-axis to the point (3, 0) and
√ then√counterclockwise around the
circumference of the circle x2 + y 2 = 9 to the point (3/ 2, 3/ 2).
2
2
!
!
!
7. Let S be the
! cylinder x + y = 9, for 0 ≤ z ≤ 3, oriented outward. Let F = −yz i + xz j.
! in two ways: directly and using Stokes’ Theorem.
Calculate S curl F! · dA
8. Find the flux of F! through the surface S, where F! = 3x!i + y!j + z!k, and S is the part of the
surface z = −2x − 4y + 1, oriented upward, with (x, y) in the triangle R with vertices (0, 0),
(0, 2), (1, 0).
9. Use the Divergence Theorem to find the flux of F! = x3!i + y 3!j + z 3!k through the closed
surface bounding the solid region x2 + y 2 ≤ 4, 0 ≤ z ≤ 5, oriented inward.