As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region — inside the hexagon but outside all of the semicircles? $\textbf {(A) } 6\sqrt{3}-3\pi \qquad$ $ \textbf {(B) } \frac{9\sqrt{3}}{2} - 2\pi\ \qquad$ $ \textbf {(C) } \frac{3\sqrt{3}}{2} - \frac{\pi}{3} \qquad$ $ \textbf {(D) } 3\sqrt{3} - \pi \qquad$ $ \textbf {(E) } \frac{9\sqrt{3}}{2} - \pi$