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I was watching the Numberphile's episode on Gabriel's Horn paradox where they separately calculated

  • the volume of the horn
  • and its surface

The volume was calculated by slicing the horn into small pieces of width $\textrm{d}x$ and assuming that for infinitesimally small $\textrm{d}x$ they can be modeled by cylinders, yielding

$$V = \pi \int_1^\infty \frac{1}{x^2} \textrm{d}x = \pi$$

For the surface, however, they did not pretend the same - the infinitesimal parts were not modeled as rings (and then follow the same path to find the area).

There was a very short handwaving mention of why looking at them as conical frustra instead is important but that was it.

My question (please note that the Gabriel's Horn paradox is just an illustration of the general question which follows): is there a general rule for when taking infinitesimal parts of [whatever] and pretending that they are [something easy to calculate] is the proper approach, and when not?

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    $\begingroup$ You should write each expression down that you're particularly interested in. We're not going to sit through a 20 minute long video. $\endgroup$ Feb 18, 2021 at 19:34
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    $\begingroup$ You have to look at the definition of the thing you are trying to calculate (e.g. volume or surface area). In this case, you are confused mainly because surface area is actually kind of tricky to define. Once you accept a definition of surface area, I expect that what they did will seem completely obvious. $\endgroup$
    – sasquires
    Feb 18, 2021 at 19:35
  • $\begingroup$ @CameronWilliams: I am not asking for that, but whether there is a rule for the last sentence of the question. $\endgroup$
    – WoJ
    Feb 18, 2021 at 19:38
  • $\begingroup$ @sasquires: the thing I am confused with is why the surface is not modeled as a ring - and consequently why there are cases when I can do that and when not. $\endgroup$
    – WoJ
    Feb 18, 2021 at 19:39
  • $\begingroup$ So you are mainly asking for a definition of surface area. Basically, why isn’t the surface area between $x$ and $x+dx$ equal to $2\pi f(x) dx$? $\endgroup$
    – sasquires
    Feb 18, 2021 at 19:44

1 Answer 1

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A general rule is "if it works in the linear case, you can use it."

Using solid cylinders for volume of revolution gives the correct answer when you apply it to a right circular cone, that is, when the bounding curve is a line.

Using the boundaries of those disks for the surface area fails because it understates the area of the right circular cone by a constant factor -- $\sec^2 \theta$ where $\theta$ is the angle between the horizontal and the bounding line.

A more sophisticated way to show this is to show that the error in using the approximation goes to zero as the fineness of partitioning the Riemann sums improves. For the volume, the error goes to zero in the limit as the partition is refined. For the surface area, the error does not go to zero, unless the disks are replaced with conical frusta.

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