\documentclass{article} \usepackage{amsmath} \begin{document} I will use the following parameterization for C: $C = \{(R\cos t, R\sin t) : -\pi < t \leq \pi\}$ In polar coordinates the region bounded by $C$ is $D = \{(r, \theta) : 0 \leq r \leq R, -\pi < \theta \leq \pi\}$ The lefthand integral evaluates to $\int_C x\,dy = \int_{-\pi}^{\pi} R\cos t \cdot R\cos t\,dt = R^2 \int_{-\pi}^{\pi} \cos^2t\,dt = \frac{R^2}{2}\int_{-\pi}^{\pi}1 + \cos2t\,dt = \frac{R^2}{2}\left[t +\frac{\sin2t}{2}\right]_{-\pi}^{\pi} = \pi R^2$ The righthand integral evaluates to$\int\int_D 1\,dx\,dy$. This is the area of a circle of radius R, that is, $\pi R^2$\end{document}