Two semicircles of radius $2$ are constructed on diameter $\overline{AB}$ of a semicircle of radius $4$. The centers of the small semicircles are located at distances $2$ and $4$ from point $A$, respectively, as shown. Both small semicircles are constructed above $\overline{AB}$. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?