Answer
$3 \pi$
Work Step by Step
Green's Theorem states that:
$\oint_CP\,dx+Q\,dy=\iint_{D}(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y})dA$
When $C$ is in the counterclockwise direction, we use: $A=\int_{C} x dy=-\int_{C} y dx$
and when $C$ is in the clockwise direction, we use: $A=-\int_{C} x dy=\int_{C} y dx$
Since the given graph is clockwise, we will use: $A=\int_{C} x dy$
or, $=\int_{C} y(t) \dfrac{dx}{dt} dt$
or, $= \int_{0}^{2 \pi} (1-\cos t)(1-\cos t) dt$
or, $= \int_{0}^{2 \pi} (1-2 \cos t+\cos^2 t) dt$
or, $= [\dfrac{3t}{2}-2 \sin t+\dfrac{\sin 2t}{4}]_0^{2 \pi}$
or, $=3 \pi$
