Answer
$$A=\frac{1}{4}(\pi+3\sqrt 3)$$
Work Step by Step
Area can be calculated as:
$$A=\int(\frac{1}{2}+cos\theta)^{2}d \theta=\int(\frac{1}{4}+cos\theta+cos^{2}\theta)d \theta$$
or, $$A=\frac{3}{4}\theta+sin\theta+\frac{1}{4}sin2\theta$$
Now, in order to find the area we will have to use the limits and we will subtract the small loop from the larger loop.
Therefore,
$A=[\frac{3}{4}\theta+sin\theta+\frac{1}{4}sin2\theta]_{0}^{2\pi/3}-[\frac{3}{4}\theta+sin\theta+\frac{1}{4}sin2\theta]_{2\pi/3}^{\pi}$
Hence, $$A=\frac{1}{4}(\pi+3\sqrt 3)$$
