How will you explain the painter's paradox in the Gabriel's horn that even if we require infinite amount of paint to cover the surface of the Gabriel's horn but finite amount of paint is sufficient to fill it with paint?
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Why paradox? In real life there is no infinity.... So such a horn cannot be constructed by the painter. – dmtri Jul 16 '20 at 06:38
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1If you're thinking about actual paint, eventually the horn is too narrow to admit any more molecules of paint, so you can't fill it with paint, either. – saulspatz Jul 16 '20 at 07:52
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There is no requirement for how thick the layer of paint on the inner surface needs to be. We can take it to be arbitrarily close to 0. Therefore, there is an infinite amount of "surface area" in even the smallest volume of paint.
For example, how many "square feet" are in 1 cubic foot? Infinite.
Annika
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You need an infinite amount of paint to cover the surface with a layer that has a constant thickness. Let's call this thickness $\epsilon > 0$.
Filling the inside of Gabriel's horn with paint uses up the same amount of paint as painting the outside of Gabriel's horn with a layer of paint whose thickness decreases proportionally to the distance from the opening of the horn. If you think of it this way, then most of the surface of the horn will be coated with a layer of paint that is way less than $\epsilon$ thick. Hence, this will use way less paint than applying a uniformly thick layer of paint.
JimmyK4542
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You model the horn using $f(x) = 1/x$, for $x \geq 1$. In general, the volume determined by the revolution of $f(x)$ around the $x$-axis, between $a$ and $b$ is $$V = \int_a^b \pi f(x)^2\,{\rm d}x,$$while the area of the surface is $$A = \int_a^b 2\pi x \sqrt{1+f'(x)^2}\,{\rm d}x.$$Well, in this case we have $$V = \int_1^{+\infty} \frac{\pi}{x^2}\,{\rm d}x < +\infty \quad\mbox{but}\quad A = \int_1^{+\infty} 2\pi x \sqrt{1+\frac{1}{x^4}}\,{\rm d}x = +\infty.$$
Ivo Terek
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