4. Consider the gray surface in the figure, bounded by semicircles, the largest of radius 2 and the two smaller ones of radius 1. Determine the volume of the region of space traversed by the surface during a $180^{\circ}$ rotation about the dashed line.
(A) $\frac{2}{3} \pi$
(B) $4 \pi$
(C) $\frac{16 \pi}{3}$
(D) $\frac{35 \pi}{6}$
(E) $6 \pi$
4. The answer is (C). We observe that the required volume is half of that of a sphere with radius 2; the solid obtained by rotation is indeed a half sphere of radius 2, from which a hemisphere of radius 1 (corresponding to the small semicircle on the left) has been removed and an identical hemisphere (corresponding to the semicircle on the right) has been added. We must therefore calculate the volume of a hemisphere of radius 2, which is $\frac{1}{2} \cdot \frac{4 \pi}{3} \cdot 2^{3}=\frac{16 \pi}{3}$.