First, calculate the length \(AQ\) which is the radius of the arc. Since \(Q\) is the midpoint of \(BC\), and \(ABC\) is an equilateral triangle, \(AQ = \sqrt{3}\).

Next, calculate the area of triangle \(AQC\). The height from \(A\) to \(BC\) is \(\sqrt{3}\), so the area of \(\triangle AQC\) is \(\frac{1}{2} \times 2 \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}\).

Now, calculate the area of the sector \(APR\). The angle \(\angle BAC\) is \(60^\circ\) or \(\frac{\pi}{3}\) radians. The area of the sector is \(\frac{1}{2} \times (\sqrt{3})^2 \times \frac{\pi}{3} = \frac{\pi}{2}\).

The area of the shaded region is the area of \(\triangle AQC\) minus the area of the sector \(APR\):

\(\text{Shaded area} = \frac{\sqrt{3}}{2} - \frac{\pi}{2} = \sqrt{3} - \frac{\pi}{2}\)