Solid of revolution vs. area below 1/x

[sweatingbear]

How come the area below the graph of \(\displaystyle \frac 1x\) between \(\displaystyle [1, \infty)\) does not exist, but the solid of revolution below the same graph in that same interval does exist? I do not see the logic.

 

[chisigma]

How come the area below the graph of \(\displaystyle \frac 1x\) between \(\displaystyle [1, \infty)\) does not exist, but the solid of revolution below the same graph in that same interval does exist? I do not see the logic.

The calculus demonstrates that the area...


$\displaystyle A = \int_{1}^{\infty} \frac{dx}{x} = \infty$ (1)

...and the volume of its the solid of revolution is finite and is...


$\displaystyle V= \pi\ \int_{1}^{\infty} \frac{dx}{x^{2}} = \pi$ (2)

It may seem incredible but this 'paradox' was discovered by Pappo di Alessandria in the fourth century after Christ [!]...


Kind regards


$\chi$ $\sigma$

 

[MarkFL]

You may find this article interesting:

Gabriel's Horn - Wikipedia, the free encyclopedia

 

[sweatingbear]

Thank you both for your replies.