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How can a structure have infinite length and infinite surface area, but have finite volume?


I have this question that I quite cant explain why.

So the area under the curve $$y=\frac{1}{x}$$ from 1 to infinity equals $$\lim_{r->\infty } \ln(r) |_1^r = \infty $$

And the volume of the solid by rotation the equation about the x-axis from 1 to infinity is

$$\int_{1}^{\infty} \pi\left (\frac{1}{x} \right )^2 dx$$ which evaluates to the area being $$\pi$$

So, whats the explanation for the area being infinite but the volume being finite?

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marked as duplicate by Qiaochu Yuan Jun 1 '11 at 22:04

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

up vote 9 down vote accepted

This is called Gabriel's horn." There isn't much to "explain": it just is, as a consequence of how things work. In short, it's a paradox (a result that is counter to expectations or intuition), not a contradiction.

You can accomplish a similar situation if you imagine starting with a cylinder of, say, radius $r$ and height $h$. The volume is $\pi r^2h$, the surface area is $2\pi rh$. If try to make the cylinder taller and thinner so as to keep the volume constant, you will necessarily increase the height a lot faster than you are decreasing the radius, because the decrease in the radius will be quadratic while the increase in the height will be linear (if you decrease the radius by a factor of $\frac{1}{4}$th, the volume decreases by a factor of $\frac{1}{16}$, so you would need to increase the height by a factor of $16$ to keep up); however, that will result in a net increase in the surface area (if you decrease the radius by $\frac{1}{4}$ and increase the height by a factor of $16$, then the surface area increases by a factor of $4$). If you keep making the cylinder taller and thinner and keeping the volume constant, then the surface area will necessarily grow without bound. So you have a "continuous deformation" of the cylinder that keeps the volume constant, but the surface area grows without bound.

With Gabriel's Horn you have a similar situation (with your figure lying on its side); intuitively, you are reducing the radius sufficiently faster than the "length" so that the total volume decreases, but not so fast that the total surface length also decreases. Intuitively, the "inside" is getting small fast enough that it does not amount to much, but just "fast enough" that the outside does not decrease fast enough.

It's just like the fact that you can add an infinite number of positive numbers, $$\frac{1}{1^2} + \frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots+\frac{1}{n^2} + \cdots$$ and yet get a finite total, but if you change it just a bit you can get something infinite: $$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} +\cdots$$ which diverges.

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