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Consider the curve $\frac{1}{x}$ where $x \geq 1$. Rotate this curve around the x-axis.

One Dimension - Clearly this structure is infinitely long.

Two Dimensions - Surface Area = $2\pi\int_∞^1\frac{1}{x}dx = 2\pi(\ln ∞ - \ln 1) = ∞$

Three Dimensions - Volume = $\pi\int_∞^1{x}^{-2}dx = \pi(-\frac{1}{∞} + \frac{1}{1}) = \pi$

So this structure has infinite length and infinite surface area. However it has finite volume, which just does not make sense.

Even more interesting, the "walls" of this structure are infinitely thin. Since the volume is finite, we could fill this structure with a finite amount of paint. To fill the structure the paint would need to cover the complete surface area of the inside of this structure. Since the "walls" are infinitely thin, why would a finite amount of paint not be able to cover the outside of the "walls" too?

Please help me make sense of this whole thing.

  • 14
    The infinite is often counterintuitive. You cannot *actually* fill it with paint, though: Planck would get in your way (eventually, the horn is thinner than atoms).2010-12-17
  • 7
    This is known as Gabriel's Horn: http://mathworld.wolfram.com/GabrielsHorn.html2010-12-17
  • 0
    Related, I suppose: http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox2010-12-17
  • 5
    A simpler $2$-dim example: one obtains finite area and infinite perimeter by appending the rectangles $\rm R_n$ of height $\rm 1/2^n $ and width $1,$ for all $\rm\ i\in \mathbb N$2010-12-17
  • 1
    Does the existence of infinitely long regions of the plane with finite area, such as the area under $y = exp(-x)$ in the first quadrant, involve any sort of paradox? The finite volume of rotation described by pacman is similarly an instance of how a finite value can be the limit of an infinite series.2010-12-17
  • 0
    To continue on the jocular connection to "real world" physics: "Planck would get in your way" - Yes, under a certain finite pressure. If we stuffed hard enough to overpower the forces deciding the shape of the atom, we could continue filling it with paint. It would put quite a stress on the walls, though, but if we make assumptions as "infinitely thin", why not also throw in "infinitely rigid"? :-) Now we have a new physical frontier to battle against.2011-02-01
  • 0
    Your integral limits are reversed and your expression for the surface area is not quite correct, and should be more like $2\pi \int_1^\infty \frac{1}{x} \sqrt{1 + 1/x^4} \; dx$, though this is still infinite.2011-06-01
  • 0
    I remember the first time I was shown Gabriel's Horn. It blew my mind :)2011-07-20
  • 0
    This really isn't as surprising as people say. Ordinary examples are all around. A kitchen sponge has a small volume but an enormous surface area; adding more holes to the sponge increases the surface area but *decreases* the volume. Notice that if you take a piece of cheese and cut it in half, the volume stays the same but the surface area increases, so by chopping it into a large number of pieces you can increase the surface area to infinity while the volume remains constant. Wikipedia reports that a gram of [activated charcoal](http://enwp.org/Activated_carbon) can have a volume of 500 m².2014-07-03

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