Volume of Revolution: Intuitive Explanation

[Aeneas]

If f(x) = x to a power between -0.5 and -1, the area between the f(x) graph and the x-axis from, say x=1 to infinity is infinite, but the volume of revolution of f(x) around the x-axis is finite. This seems counter-intuitive. Can anyone give a satisfying explanation of this - preferably a geometrical one please - not just the algebraic integration please - as I'm struggling with this idea.

Thanks, in anticipation.

 

[daveb]

What makes you say the volume is finite?

 

[ice109]

for $$\frac{1}{x}$$ he's right volume is finite but surface area and integral are infinite

http://en.wikipedia.org/wiki/Gabriel's_Horn

down at the bottom there's a good explanation

 

[Aeneas]

daveb - For -1 is less than the power of x is less than -0.5 (Sorry, I don't have any notation available) adding 1 to the power will; make that power positive, so the chosen point, say 's' above 1 will be in the numerator and the term will become infinite as 's' tends to infinity. When you square the function, though, to get the volume, the power of x will be doubled, making it less than -1, so when the result is integrated, the power of x will still be less than one, the 's' term will be in the denominator and so the term will tend to zero as 's' tends to infinity.Mnay thanks, daveb and ice109 for your replies.

 

[ice109]

daveb - For -1 is less than the power of x is less than -0.5 (Sorry, I don't have any notation available) adding 1 to the power will; make that power positive, so the chosen point, say 's' above 1 will be in the numerator and the term will become infinite as 's' tends to infinity. When you square the function, though, to get the volume, the power of x will be doubled, making it less than -1, so when the result is integrated, the power of x will still be less than one, the 's' term will be in the denominator and so the term will tend to zero as 's' tends to infinity.Mnay thanks, daveb and ice109 for your replies.

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[Aeneas]

Thanks for that tip ice109. The Gabriel's Horn explanation in Wikipedia in good, much better than Weisstein, who does not even try to tackle it, but it still does not really satisfy in terms of intuitive geometrical thinking. Also 1/x is only one of a family of functions for which this paradox occurs, which is not made clear. Is there a clue here? Thanks for that, anyway.

 

[ice109]

infinite things are not spacially intuitive, by virtue of never knowing infinity you can not have intuition about it.

 

[daveb]

Heh! That'll teach me for not actually working oiut the calculation. I guess I really deserve that genius appellation now (see https://www.physicsforums.com/showthread.php?t=175567&page=2")