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