Answer
$4 \pi$
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:
$\oint_CP\,dx+Q\,dy=- \iint_{D} \sin y -1-\sin y dA$
or, $=- \iint_{D} -1 dA$
or, $=- \int_{0}^{2\pi} \int_{0}^{2} -r \ dr \ d \theta$
or, $= - \int_{0}^{2\pi} (-2) d \theta$
or, $= 2[ 2\pi-0]$
or, $=4 \pi$
