Math 260 Exam 3 Jerry L. Kazdan
April 12, 2012 12:00 – 1:20
Directions This exam has two parts. Part A has 4 short answer questions (10 points each,
so 40 points) while Part B has 4 traditional problems (15 points each, so 60 points). Total: 100
points. Neatness counts.
Closed book, no calculators, computers, ipods, cell phones, etc – but you may use one 300 ×500 card
with notes on both sides.
Part A: Four short answer questions (10 points each, so 40 points).
A–1. Let f(x) := Zx
0Zt
0
g(s)dsdt for x≥0. Rewrite this as an iterated integral with the
order of integration reversed, so one first integrates with respect to t.
For the next 3 problems, γ(t) = (x(t), y(t)), a≤t≤b, is a smooth curve in the plane and we
consider the line integral J:= Rγp(x, y )dx +q(x, y)dy . Give a proof or counterexample for each
of the following.
A–2. If γ(t) is a horizontal line segment and p(x, y) = 0 on this segment, then J= 0 .
A–3. If γ(t) is a vertical line segment and p(x, y) = 0 on this segment, then J= 0.
A–4. If p(x, y )≥0 and q(x, y)≥0 on γ, and if in defining γwe know that dx/dt > 0 and
dy/dt > 0 , then J≥0.
Part B: Four traditional problems (15 points each, so 60 points).
B–1. Let F=yi+ (3 + 2x)j+ 2k, and γ(t) be the straight line from(0,0,0) to (1,2,−3) . Compute
Zγ
F·ds.
B–2. Let G(x) := Zb(x)
a(x)
f(t)dt, where a(x) and b(x) are smooth functions with a(x)< b(x), and
f(x) is a continuous function. Compute dG(x)/dx .
B–3. Compute ZZR2
dx dy
[4 + (x−y)2+ (x+ 2y)2]2
1
