The above diagram shows the points $R=(7,\frac{1}{7})$ and $P=(14,\frac{1}{14}),$ which lie on the curve $y=\frac{1}{x},$ point $Q = (0, \frac{1}{7})$ on the $y$-axis, and points $A = (7,0)$ and $B = (14,0)$ on the $x$-axis. If $S$ is the solid obtained by rotating the region $R$ bounded by $\overline{OQ}, \overline{QR}, \widehat{RP}, \overline{PB}, \text{ and } \overline{OB}$ about the axis of rotation $x=-7$ and the volume of $S$ is $\alpha \pi,$ what is $\alpha?$

Note: The above diagram is not drawn to scale.