Answer
$\dfrac{\pi}{8}$
Work Step by Step
The area of a shaded region can be found as: $A=\dfrac{1}{2}\int_p^q r^2 d \theta$
Here, we have $A=\int_{-\pi/4}^{\pi/4} \dfrac{1}{2}\cos^2 2 \theta d \theta$
$(2)\dfrac{1}{2} \int_{-\pi/4}^{\pi/4} (\dfrac{1+\cos 4 \theta}{2})d \theta=\dfrac{1}{2}[ \dfrac{\pi}{4}+\dfrac{ \sin 4 \theta}{\theta}]_{0}^{\pi/4} $
Thus, $A=\dfrac{\pi}{8}$
