Volume of Gabriel's Horn

From ProofWiki
Jump to navigation Jump to search

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.