Volume of Gabriel's Horn/Corollary
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Corollary to Volume of Gabriel's Horn
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 plane $x = 1$ and the portion of Gabriel's horn where $x \ge 1$.
Then:
- $V = \pi$
Proof
\(\ds V\) | \(=\) | \(\ds \pi \int_1^\infty \frac 1 {x^2} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{a \mathop \to \infty} \pi \int_1^a \frac 1 {x^2} \rd x\) | Definition of Improper Integral on Closed Interval Unbounded Above | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{a \mathop \to \infty} \pi \paren {1 - \dfrac 1 a}\) | Volume of Gabriel's Horn | |||||||||||
\(\ds \) | \(=\) | \(\ds \pi \paren {1 - 0}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \pi\) |
$\blacksquare$
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.
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.15$: Torricelli ($\text {1608}$ – $\text {1647}$)