Math 265
Professor Lieberman
December 1, 2010
PRACTICE FOURTH IN-CLASS EXAM
DIRECTIONS: This practice is only intended to show the level of difficulty for Wednesday’s exam. The topics on the actual exam may be different from those represented here.
Also, read the directions for each problem carefully; the actual exam may only ask you to
set up integrals.
1. Evaluate the line integral
∫
(x + 2y) dx + (x − 2y) dy,
C
where C is the line segment from (1, 1) to (3, −1).
2. Determine whether
e−x
j
y
is conservative. If F is conservative, find f so that F = ∇f ; if F is not conservative,
say so.
F(x, y) = −e−x ln yi +
3. Use Green’s Theorem Ito evaluate
(x2 + 4xy) dx + (2x2 + 3y) dy,
C
where C is the ellipse 9x2 + 16y 2 = 144.
4. Use the divergence theorem to evaluate
∫∫
F · n dS,
∂S
where F(x, y, z) = 2xi + 3yj + 4zk and S is the solid spherical shell 9 ≤ x2 + y 2 + z 2 ≤
25.