I have to compute the Index around a positively-oriented loop that surrounds the origin. I can use the loop given by (x(s),y(s))=(cos(s),sin(s)), 0<=s<=2*pi.

The first example is:

x' = ax, y' = by, with a>0 and b>0

The formula for the Index around the loop is:
over L, where x'=P and y'=Q

So for this first example, I get:

over L

But I COMPLETELY forget how to take integrals over regions. I tried doing a double integral, from left to right, the first from 0-->2pi and the inside one from 0-->1, and then converting the inside to polar coordinates and multiplying by r, dr d(theta), but I'm not getting anywhere. Can anyone help me out?