How to solve this line integral.

Question

I'm given $$\oint_C \frac{-y}{x^2 + y^2} \,dx + \frac{x}{x^2 + y^2} \,dy $$

and C = {(x,y) : $ x^2 + y^2 = 1$

I found the integrals separately and got $$ -arctan\frac{x}{y} + arctan\frac{y}{x}$$

Now I do not how to include $x^2 + y^2 =1$ , in this case, to fully solve the integral.

Any suggestions?

Answer 1

You can parametrize it using

• $x(t) = \cos t, y(t) = \sin t$. So your integral becomes $$\oint_C \frac{-y}{x^2 + y^2} \,dx + \frac{x}{x^2 + y^2} \,dy $$ $$= \int_0^{2\pi}\binom{-\sin t}{\cos t}\cdot \binom{-\sin t}{\cos t}dt$$ $$= \int_0^{2\pi}1\;dt = 2\pi$$