Written homework problem 17
Assigned 11/5 and due 11/10
(Variation on problem 2 from the spring 2011 take-home exam 3.)
Evaluate
Z
π/2
dt
p
.
2 sin(t) cos(t)(sin(t) + cos(t))
0
[ Warning: I think if you plug this into Maple, Maple will give you the wrong answer. Here are
some hints. ]
I=
y
x
4
4
(a) Let F~ (x, y) = h− x2 +y
= 1 in the 1st
2 , x2 +y 2 i. Let C1 be the part of the curve x + y
quadrant,
p so the start point is (1, 0) and the end
p oriented
R point is (0, 1). Parametrize C1 by
~r(t) = h cos(t), sin(t)i, 0 ≤ t ≤ π/2. Show that I = C1 F~ · d~r.
(b) Let C2 be the part of the unit circle in the 1st quadrant, oriented so the start point is (1, 0)
and the end point is (0, 1). Evaluate
Z
F~ · d~r.
C2
(c) Explain why
R
C1
F~ · d~r =
R
C2
F~ · d~r.
1