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Torricelli's trumpet wrong?

  • 31-01-2006 5:44pm
    #1
    Fallen Seraph
    Registered Users, Registered Users 2 Posts: 443 ✭✭


    According to wikipedia torricelle's trumpet has infinite surface volume. Is there some proof for its infinite volume, because I'm pretty sure I could show that it has finite volume... Or has that paradox already been solved and wikipedia just doesn't have it written?


Comments

  • #2
    i71jskz5xu42pb
    Closed Accounts Posts: 1,806 ✭✭✭i71jskz5xu42pb


    Fallen Seraph wrote:
    According to wikipedia torricelle's trumpet has infinite surface volume.

    You're confused I think. The paradox is that the shape has a finite volume with an infinite surface area.

    The proof is right there in Wikipedia (or Mathworld if you want something more detailed).


  • #3
    Fallen Seraph
    Registered Users, Registered Users 2 Posts: 443 ✭✭Fallen Seraph


    Sorry, I mangled my words. I meant infinite surface area. And I know the proof of its volume, but I say that I believe that I can show that the surface area is also finite. Am I just wrong?


  • #4
    ecksor
    Moderators, Social & Fun Moderators Posts: 10,501 Mod ✭✭✭✭ecksor


    Have you attempted the proof? Or have you attempted to find a flaw in the proof presented on those pages linked by PaschalNee? That would probably be a better way to get to the heart of the matter faster.


  • #5
    Fallen Seraph
    Registered Users, Registered Users 2 Posts: 443 ✭✭Fallen Seraph


    Hmmm... Ah yes. Well would someone then mind telling me what's wrong with this analysis of the surface area of't please?


    Take any section of the 2-d graph between n and n+1 where n is x coordinate and E n and assume that it's a straight line of m=0 through y=f(n) between these points. If this is rotated about the y - axis it will form a cylinder which will have a total surface area larger than the corresponding bit of the trumpet. The TSA of a cylinder will be 2(pi)rh. h will be 1. r will be f(n). As n approaches infinity, f(n) will tend towards 0 => the TSA of the cylinder will tend towards 0. Therefore the sum of all these cylinders will approach a definite number. And since the TSA of the trumpet is less than the cylinders it must also be a definite number.


    I can't see the flaw in that reasoning, I'd greatly appreciate't if someone would be so kind as to enlighten me.


  • #6
    Fallen Seraph
    Registered Users, Registered Users 2 Posts: 443 ✭✭Fallen Seraph


    Ah, nevermind. I've seen the flaw.


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  • #7
    Capt'n Midnight
    Moderators, Recreation & Hobbies Moderators, Science, Health & Environment Moderators, Technology & Internet Moderators Posts: 93,583 Mod ✭✭✭✭Capt'n Midnight


    Tip
    If you want to paint the outside, simply half-fill a larger trumpet with paint and dip the first one inside.


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