Answer
$5\pi-8 $
Work Step by Step
The area of between the polar curve is given by as follows:
$A=(\dfrac{1}{2})\int_p^q [(r_2(\theta))^2-(r_1(\theta))^2] d \theta$
$\implies A=(\dfrac{1}{2})\int_{(-\pi/2)}^{(\pi/2)} [2^2-(2-2\cos \theta)^2] d\theta $
$\implies A=(\dfrac{1}{2})\int_{-\pi/2}^{\pi/2} [8 \cos \theta-4 \cos^2 \theta] d\theta$
This implies that
$A=\dfrac{1}{2}[8 \sin \theta-4 (1/2)\sin 2 \theta/2+\theta]_{-\pi/2}^{\pi/2}$
Thus, $A=\pi(2)^2-(8-\pi)=5\pi-8 $
