C Roettger, Fall 14
Math 265 Quiz 11B – Solutions
Problem 1 Find the line integral
Z
(4x3 − 2xy) dx + (sin y − x2 ) dy
C
where C is the path consisting of line segments from (0, 0) to (2, 0), from
(2, 0) to (2, 4), and (2, 4) to (0, π).
Solution. First, we find a potential function for the vector field
F = (4x3 − 2xy)i + (sin y − x2 )j.
It’s fairly easy to find that
f (x, y) = x4 − x2 y − cos(y)
is a potential for F (it satisfies ∇f = F ). By the Fundamental Theorem of
Calculus, the line integral in question is
Z
(4x3 − 2xy) dx + (sin y − x2 ) dy = f (0, π) − f (0, 0) = 1 − (−1) = 2.
C
If you did this for every line segment separately, you would get
f (0, π) − f (2, 4) + f (2, 4) − f (2, 0) + f (2, 0) − f (0, 0)
and the intermediate points will cancel out, giving the same answer. Another
way to see that they will cancel out is to note that F is conservative. So
it does not matter whether you travel from (0, 0) to (0, π) via three line
segments, or directly on a straight line.