We have the following curve integral:
$\int_\phi \frac{-ydx+xdy}{x^2+y^2}$
Where the curve $\phi$ is the curve around the following area, in positive direction:
We do not know the equation of the curve and we do not know which points it goes through. How do you solve this?
The curve defined by $\phi$ is clearly diffeomorphic to the unit circle of $\mathbb{R}^2$ and the differential form $\frac{x\mathrm{d}y-y\mathrm{d}x}{x^2+y^2}$ is the angle-form on $\mathbb{R}^2\setminus\{0\}$, hence the desired integral is equal to $2\pi$.