1. Evaluate the integrals
and
2. Let be the vertices of a closed quadrilateral such that traverses in counterclockwise direction (abusing notation, we shall write for that curve) show that is 0 for (C-minus Box) and 1 if is an interior point of (Hint: do not attempt to parametrize ..)
If is outside the rectange then computing the winding number is to evaluate by Cauchy's theorem. If is inside then it can be shown, here (post #8 with long proof), that if is a tiny circle around still inside then . And so the winding number is .