jmc

The area of the shaded region equals the area of the circle minus the area of the triangle. The area of the circle is $2^2\pi =4\pi$. To find the area of the triangle, we look for information about the triangle. Because angle $ACB$ intersects $180^{\circ }$ of the circle, we know $m\angle ACB=\frac{180^{\circ }}{2}=90^{\circ }$, so triangle $ACB$ is a right triangle. What's more, since $AB=4$ and $BC=2$, it follows that $AC=2\sqrt{3}$ and the area of right triangle $ACB$ equals $\frac{2\cdot 2\sqrt{3}}{2}=2\sqrt{3}$. So, the area of the shaded region is $\boxed{4\pi -2\sqrt{3}}$.