The large circle has diameter $\text{AC}$.  The two small circles have their centers on $\text{AC}$ and just touch at $\text{O}$, the center of the large circle.  If each small circle has radius $1$, what is the value of the ratio of the area of the shaded region to the area of one of the small circles?

$\text{(A)}\ \text{between }\frac{1}{2}\text{ and 1} \qquad \text{(B)}\ 1 \qquad \text{(C)}\ \text{between 1 and }\frac{3}{2}$
$\text{(D)}\ \text{between }\frac{3}{2}\text{ and 2} \qquad \text{(E)}\ \text{cannot be determined from the information given}$
The small circle has radius $1$, thus its area is $\pi$.
The large circle has radius $2$, thus its area is $4\pi$. 
The area of the semicircle above $AC$ is then $2\pi$.
The part that is not shaded are two small semicircles. Together, these form one small circle, hence their total area is $\pi$. This means that the area of the shaded part is $2\pi-\pi=\pi$. 
This is equal to the area of a small circle, hence the correct answer is $\boxed{\text{(B)}\ 1}$.