The Paradox of Gabriel’s Horn

We previously discussed the importance of partial differentiation in mathematics and physics. This post will focus particularly on using calculus to find the surface area of geometric shapes.

The concept of the paradox of Gabriel’s horn or Torricelli’s trumpet is very popular and interesting to many mathematicians. The horn object is created using the function of the inverse of x in the domain 1 ≤ x ≤ ∞ spun about the x axis. The domain is chosen to avoid the asymptote at x = 0.

     \[       f(x)= {\frac{1}{x}}   \]

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Whats interesting about this shape, as we will see, is that it has an infinite surface area yet has a finite volume.

If we consider the horn to be a container full of paint we can easily visualize why this concept is considered a paradox. It is impossible to imagine a container that can hold a finite amount of paint yet require an infinite amount of paint to coat it’s inside wall.

For this example we will use calculus to prove that this peculiar object has an infinite surface area yet has a finite volume.

First we’ll find the volume using simple integration.

     \begin{align*}     V & =  \pi \int_1^\infty{ f(x)^2 } \,\mathrm{d}x \\ & =   \pi \int_1^\infty{ \frac{1}{x^2}  } \,\mathrm{d}x \\ & = \lim_{n \to \infty}  \pi \int_1^n{ \frac{1}{x^2}  } \,\mathrm{d}x \\  &=  \lim_{n \to \infty}  \pi \left [ { - \frac{1}{x} } \right ] _ 1 ^ n  \\ &= \pi \left [  { \left ( - \frac{1}{(\frac{1}{0})} \right ) -  \left ( -\frac{1}{(1)} \right )   } \right ] \\ &= \pi (  { 0 + 1 } ) \\ &= \pi  \\ \end{align*}

We have just proven that the volume of the horn is π and therefore is finite.

In the last post I introduced the concept of finding the surface area of 3-dimensional objects using a combination of integration and partial differentiation. The previous post introduced a general equation that can be used. However for this example we’re going to use an easier equation to determine the arc length of a function and simply spin it about the x-axis.

     \begin{align*}   SA &= \int_1^{\infty}{  2\pi f(x) \sqrt{ 1 + \left [  \frac{\partial y}{ \partial x} \right ] ^ 2 }  } \,\mathrm{d}x \\ &= \int_1^{\infty}{  2\pi \left ( \frac{1}{x} \right )  \sqrt{ 1 + \left (   - \frac{1}{ x^2 } \right ) ^ 2 }  } \,\mathrm{d}x \\ &= 2\pi \int_1^{\infty}{  \frac{1}{x}   \sqrt{ 1 + \frac{1}{ x^4 }  }  } \,\mathrm{d}x \\ \end{align*}

Fortunately we don’t actually need to solve this integral since the following is true.

     \[     \sqrt{ 1 + \frac{1}{ x^4 }  }  > \sqrt{1} = 1 \]

Therefore it is obvious that the following is true as well.

     \begin{align*}    2\pi \int_1^{\infty}{  \frac{1}{x}   \sqrt{ 1 + \frac{1}{ x^4 }  }  } \,\mathrm{d}x  &>    2\pi \int_1^{\infty}{  \frac{1}{x}   \sqrt{ 1 }  } \,\mathrm{d}x &=   2\pi \int_1^{\infty}{  \frac{1}{x}  } \,\mathrm{d}x \\ \end{align*}

Using this trick we can predict how the integral converges and we can now simply solve the following.

     \[  2\pi \int_1^{\infty}{  \frac{1}{x}  } \,\mathrm{d}x \\ \] \begin{align*}   &=  2\pi \lim_{n \to \infty} \int_1^{n}{  \frac{1}{x}  } \,\mathrm{d}x \\ &=  2\pi \lim_{n \to \infty} \left [ ln(x) \right ] _1^n\\ &=  2\pi \left [ \left ( ln(\infty) \right)  -  \left ( ln(1) \right)   \right]\\ &=  2\pi \left [ \left ( ln(\infty) \right)  -  (0)   \right]\\ &= \infty\\ \end{align*}

So it can be shown that Gabriel’s horn has an infinite surface area yet has a finite volume.

The mystery behind the paradox can easily be explained however. The volume is finite because of the way the 3-dimensional units of volume converge as x approaches infinity.

The posts so far have focused primarily on integral calculus and differentiation. In the next few posts I will begin to discuss linear algebra and introduce the use of matrices as well as discuss the importance of eigenvalues and eigenvectors.

Dan Roy, Founder and Creator of Universal Physics.

Category(s): Mathematics
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