AMC 12 2003a Problem 15

A semicircle of diameter $1$ sits at the top of a semicircle of diameter $2$, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a ''lune''. Determine the area of this lune.

Image:2003amc10a19.gif

$\mathrm{(A) \ } \frac{1}{6}\pi-\frac{\sqrt{3}}{4} \mathrm{(B) \ } \frac{\sqrt{3}}{4}-\frac{1}{12}\pi \mathrm{(C) \ } \frac{\sqrt{3}}{4}-\frac{1}{24}\pi \mathrm{(D) \ } \frac{\sqrt{3}}{4}+\frac{1}{24}\pi \mathrm{(E) \ } \frac{\sqrt{3}}{4}+\frac{1}{12}\pi$