A semicircle of diameter $2\sqrt{2}$ sits at the top of a semicircle of diameter $4$. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune.

1. The smaller semicircle has radius $\sqrt2$, so its area is $\frac12\pi(\sqrt2)^2=\pi$.
2. The radii from the large semicircle's center to the two intersection points are perpendicular. The overlap is a $90^\circ$ sector of radius $2$ minus a right triangle with legs $2$, so its area is $\frac14\pi(2)^2-\frac12(2)(2)=\pi-2$.
3. The lune's area is $\pi-(\pi-2)=\boxed{2}$.