The radius of the semicircle is \(\frac{1}{2} AB = r \sin \theta\).

The area of the semicircle is \(\frac{1}{2} \pi r^2 \sin^2 \theta = A_1\).

The shaded area is the area of the semicircle minus the area of the segment.

Shaded area = \(A_1 - \frac{1}{2} r^2 \theta + \frac{1}{2} r^2 \sin 2\theta\).