Analysis Exam Questions I am going to have an analysis exam soon and I found the following question in a past paper:
Evaluate
$$\int \frac{-y \, dx + x \, dy}{x^2+y^2}$$
a) Once counterclockwise around the circle $$x^2 + y^2 = 1$$
b) Once counterclockwise around the circle $$(x-2)^2 + y^2 = 1$$
Could someone please tell me how to solve this? I can't seem to get my head around as to how I should start.

Parameterize the contour.
a) $x=\cos{t}$, $y=\sin{t}$, $dx=-\sin{t} \, dt$, $dy=\cos{t} \, dt$, $t \in [0,2\pi)$
$$\oint_{\{(x,y):x^2+y^2=1\}} \frac{-y \, dx + x \, dy}{x^2+y^2} = \int_0^{2 \pi} dt \frac{\cos^2{t}+\sin^2{t}}{\cos^2{t}+\sin^2{t}} = 2 \pi$$
Can you do b)?  (This time, $x=2 + \cos{t}$,...)
