How to evaluate a line integral when the curve is oriented clockwise I am trying to calculate the line integral $\int_{C}^{} \frac{-y}{x^2+y^2}dx +\frac{x}{x^2+y^2}dy$ where C is any circle centered at the origin with a clockwise orientation.
I can set up the integral fine. My problem is with the orientation of the curve.
If the curve is oriented clockwise the shouldn't the bounds of the integral for the paramterisation go like $\int_{0}^{-2\pi}$? Because that would mean we start and $\theta = 0$ and then travel clockwise around until we reach $\theta = 0$ again. Also is this equivalent to $\int_{-\pi}^{\pi}$?

You could make the ordinary substitution $x=r\cos\theta$ and $y = r \sin \theta$ with $\theta$, as you said, running the limits backwards from $0$ to $-2\pi$.
$$\int_0^{-2\pi}$$
Alternatively, you could make the substitution $x=r\cos(-\theta) \equiv r\cos\theta$ and $y=r\sin(-\theta) \equiv -r\sin\theta$ and run the limits as normal, i.e. from $0$ to $2\pi$.
$$\int_0^{2\pi}$$
