Infinite area under a curve has finite volume of revolution? 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.

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.
