Volume of Gabriel's Horn
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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:
- $V = \pi \paren {1 - \dfrac 1 a}$
Corollary
Consider the volume $V$ of the space enclosed by the plane $x = 1$ and the portion of Gabriel's horn where $x \ge 1$.
Then:
- $V = \pi$
Proof
From Volume of Solid of Revolution:
\(\ds V\) | \(=\) | \(\ds \pi \int_1^a \frac 1 {x^2} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \pi \intlimits {-\dfrac 1 x} 1 a\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \pi \intlimits {\dfrac 1 x} a 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \pi \paren {1 - \dfrac 1 a}\) |
$\blacksquare$
Historical Note
The volume of Gabriel's horn was first demonstrated by Evangelista Torricelli in $1643$.
The result caused great astonishment at the time, as it was the first example of a solid figure demonstrated to be of infinite extent while also having finite volume.
As Thomas Hobbes put it:
- To understand this for sense, it is not required that a man should be a geometrician or a logician, but that he should be mad.