Gabriel's Horn

Both Finite and Infinite

When we study Improper Integration, it's common to encounter areas that extend infinitely far left or right, but quantify to a finite result. One such popular example is$$\int_{1}^{\infty} \frac{1}{x^2} \,\, dx$$Which, as you can prove for yourself with what we learned about improper integrals, evaluates to $1$. A graph of the function and the enclosed area is presented below for reference.

We also study volumes of revolution in Calculus by taking graphs of the form $f(x)$ and rotating them around the $x$-axis. The most common method, known as the Disks and Washers » method, explains that we can create a volume by rotating around each cross-slice of the function to create a thin disk, and then integrate each disk's volume ($\pi r^2$ where $r$ is really the function height $f(x)$) to get the volume.Here's a visual example of the volume of revolution of the function $f(x) = \sqrt{x}$ from $x=0$ to $x=4$.As we learned in the lesson, the volume of revolution of function $f(x)$ from $x=a$ to $x=b$ is given by the equation$$\pi \, \int_{a}^{b} [f(x)]^2 \,\, dx$$In the Geogebra example above, we could quickly calculate the volume that is created when $f(x) = \sqrt{x}$ is rotated around the $x$-axis from $x=0$ to $x=4$.$$\pi \int_{0}^{4} (\sqrt{x})^2 \,\, dx$$$$=\pi \int_{0}^{4} x \,\, dx$$$$=\pi \left[ \frac{x^2}{2} \right]_{0}^{4}$$$$=8\pi$$Soon after we learn about volumes of revolution, it's typical to learn how to calculate surface areas of revolution ». Surface areas of revolution are visually similar to volumes in terms of how we get from a normal $f(x)$ function to a rotated 3D shape, though the formula is a little more complicated because it uses function arc length, which is itself more complicated than area and volume based formulas.The formula for a surface area of revolution of a function $f(x)$ from $x=a$ to $x=b$ is$$2\pi \int_{a}^{b} f(x) \sqrt{1+\left( f'(x)\right)^2} \,\, dx$$Now we'll look at the fabled Gabriel's Horn object."Gabriel's Horn", which sometimes goes by the name "Torricelli's Trumpet", is a figure of mathematical intrigue because it has finite volume but infinite surface area. The name "Gabriel's Horn" is said to come from those who see it as a link between the finite and the infinite, and some believe it is the angel Gabriel who blows the trumpet signaling the resurrection of the dead as described in the Bible. Although this object is more popularly known as Gabriel's Horn, Torricelli was the one who is credited with discovering it.The mathematical definition of Gabriel's Horn is the volume of revolution created by the function $f(x) = 1/x$ from $x=1$ to $x=\infty$.

Using our two Calculus formulas, we can quickly show why this object does indeed have finite volume but infinite surface area.

Volume

$$\pi \, \int_{a}^{b} [f(x)]^2 \,\, dx$$$$\pi \, \int_{1}^{\infty} \left(\frac{1}{x}\right)^2 \,\, dx$$$$=\pi \lim_{R \to \infty} \int_{1}^{R} \frac{1}{x^2} \,\, dx$$$$=\pi \lim_{R \to \infty} \left[ -\frac{1}{x} \right]_{1}^{R}$$$$=\pi \lim_{R \to \infty} \left(1 - \frac{1}{R}\right)$$$$=\pi(1-0)=\pi$$

Surface Area

$$2\pi \int_{a}^{b} f(x) \sqrt{1+\left( f'(x)\right)^2} \,\, dx$$$$2\pi \int_{1}^{\infty} \frac{1}{x} \sqrt{1+\left( -\frac{1}{x^2} \right)^2} \,\, dx$$$$=2\pi \lim_{R \to \infty} \int_{1}^{R} \frac{1}{x} \sqrt{1 + \frac{1}{x^4}} \,\, dx$$This integral is actually pretty damn tough to work out, but we can prove its divergence via comparison. Since $x$ only takes on positive values, we can say:$$\frac{1}{x} \sqrt{1 + \frac{1}{x^4}} > \frac{1}{x}$$Therefore,$$\int_{1}^{R} \frac{1}{x} \sqrt{1 + \frac{1}{x^4}} \,\, dx > \int_{1}^{R} \frac{1}{x} \,\, dx$$$$\int_{1}^{R} \frac{1}{x} \sqrt{1 + \frac{1}{x^4}} \,\, dx > \ln(R) $$Taking the limit as $R \to \infty$ of both sides proves that$$=2\pi \lim_{R \to \infty} \int_{1}^{R} \frac{1}{x} \sqrt{1 + \frac{1}{x^4}} \,\, dx > \infty$$You can actually do the integral properly but it's quite a bit complicated, and involves inverse hyperbolic trig functions. No thanks.

Inherent Paradoxes

Gabriel's Horn is often referred to in the context of discussing paradoxes. Many functions have finite area under the curve with infinite bounds, but $f(x) = 1/x$ is special because its area from $x=1$ to $x=\infty$ does not converge to a finite amount, but its volume does. Furthermore, it is the only non-fractal 3D shape (as far as I can tell) that has finite volume and infinite surface area. These two properties teeter on the edge of the mystic, where finite meets infinite.It is also referred to when discussing what they call the "Painter's Paradox". Stemming from the mathematical properties that Gabriel's Horn has, the paradox is that the horn can be filled with a finite amount of paint, but that paint would not be enough to coat the inside of the horn where the paint was being held. This truly seems paradoxical.