Answer
$\dfrac{-16}{3}$
Work Step by Step
Green's Theorem states that:
$\oint_C M \,dx+ N \,dy=\iint_{D}(\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y})dA$
We will use polar coordinates as follows: $x=r \cos \theta; y= r \sin \theta$
We need to work out with the line integral and evaluate the integrand of the double integral as follows:
Consider $I=\iint_{D} y-x \sin x +\cos x -(\cos x -x \sin x) \ dA$
$= \int_{0}^{2} \int_{0}^{-2x+4} \ y \ dy \ dx$
$=\dfrac{1}{2} \int_{0}^{2} [y^2]_{0}^{-2x+4} \ y \ dy \ dx$
$=\dfrac{1}{2} \int_{0}^{2} (4x^2-16x +16) dx$
$=2[\dfrac{1}{3} x^3-2x^2+4x]_0^2$
$=\dfrac{-16}{3}$
