Volume of Gabriel's Horn

Theorem
Consider Gabriel's horn, the solid of revolution formed by rotating about the $x$-axis the curve:


 * $y = \dfrac 1 x$

Consider the volume $V$ of the space enclosed by the planes $x = 1$, $x = a$ and the portion of Gabriel's horn where $1 \le x \le a$.

Then:
 * $V = \pi \paren {1 - \dfrac 1 a}$

Proof
From Volume of Solid of Revolution: