Answer
$3 \pi$
Work Step by Step
When $C$ is in a counterclockwise direction, then we use: $A=\int_{C} x dy=-\int_{C} y dx$
and when $C$ is in a clockwise direction, then we use: $A=-\int_{C} x dy=\int_{C} y dx$
Green's Theorem states that:
$\oint_C A\,dx+B \,dy=\iint_{D}(\dfrac{\partial B}{\partial x}-\dfrac{\partial A}{\partial y}) dA $
We need to set up the line integral and compute the integrand of the double integral as follows:
Since the given graph is clockwise, then we will set up the integral as:
$$A=\int_{C} x dy\\=\int_{C} y(t) \dfrac{dx}{dt} dt \\= \int_{0}^{2 \pi} (1-\cos t)(1-\cos t) dt \\= \int_{0}^{2 \pi} (1-2 \cos t+\cos^2 t) dt \\= [\dfrac{3t}{2}-2 \sin t+\dfrac{\sin 2t}{4}]_0^{2 \pi}\\ =\dfrac{3(2\pi-0)}{2}-2 (\sin 2 \pi- \sin 0)+\dfrac{1}{4}(\sin 2 \pi- \sin 0) \\=3 \pi$$
