Your computation seems correct. In fact, it's more correct than I would have done. True confessions: I just looked it up, and the positive orientation of a contour is counterclockwise (the interior of the region is to your left as you traverse in the positive direction). So forget what I said about the minus sign.

The ML estimate looks like this: suppose

is a piecewise smooth curve. If

is a continuous function on

, then

.

Further, if

has length

, and

on

, then

.

That's the theorem on page 105 of Gamelin. This estimate might be useful for your

, but not for

. Why, might you ask, is it not useful for

? Because you want to know its exact value! However, if you can prove that the integral over the

is zero using the ML estimate, then you're essentially done. How could you use the ML estimate to show that the

integral is zero (as I think you'll find it is, after you've taken the limit as

)?