Vector Calculus · Superchallenge problem

Evaluate $\displaystyle\lim_{r\to 0^+}\dfrac{1}{r^4}\oint_{C_r} 2\sin y\,dx + \sin 2x\,dy$, where $C_r$ is the circle $x^2 + y^2 = r^2$ traversed once counterclockwise. Give your answer in terms of $\pi$.