In the first quadrant; the region between the x-axis, the hyperbola $x ^ { 2 } - y ^ { 2 } = 1$, and the circle $x ^ { 2 } + y ^ { 2 } = 7$ is rotated $360 ^ { \circ }$ around the x-axis.
Which of the following is the integral expression of the volume of the solid of revolution obtained?
A) $\pi \int _ { 1 } ^ { 2 } \left( x ^ { 2 } - 1 \right) d x + \pi \int _ { 2 } ^ { \sqrt { 7 } } \left( 7 - x ^ { 2 } \right) d x$
B) $\pi \int _ { 1 } ^ { 2 } \left( x ^ { 2 } + 1 \right) d x + \pi \int _ { 2 } ^ { \sqrt { 7 } } \left( 7 + x ^ { 2 } \right) d x$
C) $\pi \int _ { 1 } ^ { 2 } \left( x ^ { 2 } - 1 \right) d x + \pi \int _ { 2 } ^ { \sqrt { 7 } } ( 7 - x ) ^ { 2 } d x$
D) $\pi \int _ { 1 } ^ { \sqrt { 3 } } ( x - 1 ) ^ { 2 } d x + \pi \int _ { \sqrt { 3 } } ^ { \sqrt { 7 } } \left( 7 + x ^ { 2 } \right) d x$
E) $\pi \int _ { 1 } ^ { \sqrt { 3 } } ( x - 1 ) ^ { 2 } d x + \pi \int _ { \sqrt { 3 } } ^ { \sqrt { 7 } } ( 7 - x ) ^ { 2 } d x$