Answer
$\dfrac{\pi}{2}$
Work Step by Step
Green's Theorem states that:
$\oint_C A\,dx+B \,dy=\iint_{D}(\dfrac{\partial B}{\partial x}-\dfrac{\partial A}{\partial y}) dA $
We need to set up the line integral and compute the integrand of the double integral as follows:
$$\iint_{D} (2y-2x) dA= \int_{-\pi/2}^{\pi/2} \int_{0}^{\cos x} (2y-2x) \ dy \ dx\\= \int_{-\pi/2}^{\pi/2} [y^2-2xy]_{0}^{\cos x} \ dx\\= \int_{-\pi/2}^{\pi/2} \cos^2 x dx\\=2 \int_{0}^{\pi/2} \dfrac{1+\cos 2 x}{2} \ dx \\=[x+\dfrac{\sin 2x}{2}]_0^{\pi/2} \\=\dfrac{\pi}{2}$$
