Denote the circle of radius by . is simple and closed, so by Green's theorem the line integral reduces to a double integral over the interior of (call it ):
is a circle of radius , so we can write the double integral in polar coordinates as
a. For , we have
b. Let denote the integral with unknown parameter ,
By the fundamental theorem of calculus,
has critical points when
If , then line integral is 0, so we ignore that critical point. For the other two, we would find .