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So I was thinking about the harmonic series, and how it diverges, even though every subsequent term tends toward zero. That meant that its integral from 1 to infinity should also diverge, but would the volume of revolution also diverge (for the function y=1/x)? I quickly realized that its volume is actually finite, because to find the volume of revolution the function being integrated has to be squared, which would give 1/x^2, and, as we all know, that converges. So, my question is, are there other functions that share this property? The only family of functions that I know that satisfy this is 1/x, 2/x, 3/x, etc.

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Two related questions. –  J. M. May 12 '11 at 2:56
    
Unfortunately, I cannot find a better link for reference than this link, but "Infinite Acres" is a rather old MAA animated short film (part of a larger series, as in the link) about a solid of revolution where the original region has infinite area, the solid has finite volume, and the solid has infinite surface area. –  Isaac May 12 '11 at 4:19

3 Answers 3

up vote 5 down vote accepted

$\frac{1}{x^p}$ with $\frac{1}{2} < p \leq 1$ all satisfy these properties.

Then, by limit comparison test, any positive function $f(x)$ with the propery that there exists a $\frac{1}{2} < p \leq 1$ so that

$$ \lim_{x \to \infty} x^p f(x) = C \in (0, \infty) \,.$$

also has this property... This allows you create lots and lost of example, just add to $\frac{\alpha}{x^p}$ any "smaller" function. (i.e. $o(\frac{1}{x^p} )$)

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User's answer is a good one - but I wanted to mention a related topic. You might also note that the surface area of your object is also infinite, despite its finite volume. Thus, if you were to 'hold' such an object, you could fill it with paint but never cover its walls. This has a name - it's Gabriel's Horn ( or Torricelli's Trumpet), and you can read about it here.

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Well played –  Nicolas Villanueva May 12 '11 at 3:25

To add to the above, anything of the form $y=\dfrac{a(\ln x)^n}{x}$ where $a\neq 0$ shares this feature (the family of functions you mention being the case $n=0$)

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