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