Gabriel's Horn
Category: Calculus Stuff »
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.
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.- Popular Content Misfit Math »