Answer
$\dfrac{\pi+2}{4}$
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 $A=\dfrac{1}{2}\int_{\pi/4}^{\pi/2} (2 \sin \theta)^2 d \theta$
$\dfrac{4}{2}\int_{\pi/4}^{\pi/2}[\dfrac{1-\cos 2 \theta}{2}] d \theta=[\theta-\dfrac{1-\sin 2 \theta}{2}]_{\pi/4}^{\pi/2}$
Thus, $A=(\dfrac{\pi}{2}-0)-(\dfrac{\pi}{4}-\dfrac{1}{2})=\dfrac{\pi+2}{4}$
