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 Question from Gary: I was reading about torricelli's trumpet which is described by the equation1/x which is then rotated around the x axis which results in a figure which looks like a trumpet. Now in order to find the volume the integral 1/x^2 dx is used which diverges when integrated so the volume is finite.However if you integrate 1/x dx which is the formuls on the plane the answer diverges. Now if you took an infinite area then rotated it around the x axis should'nt you get an infinite volume? Notice the area I am talking about is under the line 1/x not the surface area of the trumpet which is what the painters paradox is about What am I missing? Thanks

Gary,

That would be "converges when integrated" but your conclusion is correct.

However if you integrate 1/x dx which is the formuls on the plane the answer diverges.
"Now if you took an infinite area then rotated it around the x axis should'nt you get an infinite volume?"

Obviously not, as you have just proved!

Why? The closer an area element is to the axis the less volume it generated upon rotation. The further out you go along the x axis, the closer 1/x gets. So "most of" the infinite area is not able to generate very much volume.

Note also that the area of the trumpet is infinite! So it would take an infinite amount of paint to paint the inside, but to fill it completely with paint would only take a finite amount... The resolution of this apparent paradox is that we were assuming a coat of paint of constant thickness, which would soon be too thick to fit into the interior.

One of the reasons why this is so counterintuitive is that the integral of 1/x goes to infinity very slowly - we do not have good intuition for logarithmic-order things.

Good Hunting!
RD

Gary wrote back

Thank-you for the reply and yes it converges not diverges as I mistakenly wrote. I noticed my mistake as soon as I had sent the letter.Here is another observation about that problem.Supposing we take the 1/x curve and it's area then lift it off the page and into the third dimension and keep that height constant then we will have a volume which I assume will be infinite.However if that height decreases as 1/x then the volume will be converging from 2 directions and therefore will reach a limit.Thanks again for the answer I think I have got it now I suppose a somewhat related question is if the 1/x graph extended into the 3'rd dimension with constant height has infinite volume would it really be possible to pack the universe as we know it into this figure?I'm having a little trouble with that! Gary

Yes, but obviously not without deforming it.

RD

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