As shown in the figure, there is a line $l : x - y - 1 = 0$ and a hyperbola $C : x ^ { 2 } - 2 y ^ { 2 } = 1$ with one focus at point $\mathrm { F } ( c , 0 )$ (where $c < 0$).
When the region enclosed by the line $l$ and the hyperbola $C$ is rotated about the $y$-axis, what is the volume of the solid of revolution? [3 points]\\
(1) $\frac { 5 } { 3 } \pi$\\
(2) $\frac { 3 } { 2 } \pi$\\
(3) $\frac { 4 } { 3 } \pi$\\
(4) $\frac { 7 } { 6 } \pi$\\
(5) $\pi$