Answer
$3 \pi$
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:
When $C$ is in counterclockwise direction then $A=\int_{C} x dy=-\int_{C} y dx$ and when $C$ is in clockwise direction then $A=-\int_{C} x dy=\int_{C} y dx$
We can see that the graph is clockwise, then we have: $$A=\int_{C} x dy = \int_{C} y(t) \dfrac{dx}{dt} dt \\= \int_{0}^{2 \pi} (1-\cos t)(1-\cos t) dt = \int_{0}^{2 \pi} (1-2 \cos t+\cos^2 t) dt \\= [\dfrac{3t}{2}-2 \sin t+\dfrac{1}{4} \sin 2t]_0^{2 \pi} \\=3 \pi$$
