smc

Let the length $AB$ be $L$. Then, we see that the region is just the union of the cylinder with central axis $\overline{AB}$ and radius $3$ and the two hemispheres connected to each face of the cylinder (also with radius $3$). Thus the volume is $9\pi L+\frac{4}{3}\pi (3{)}^3=9\pi L+36\pi (=216\pi )$ $9\pi L=180\pi$ $L=\boxed{20}$