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Gabriel's Horn (Area and Volume)

  1. Jan 29, 2012 #1
    1. The problem statement, all variables and given/known data
    To calculate the area and volume of Gabriel's Horn between [ 1, infinity ).
    And at the same time prove that, volume closes to finity, while area (or surface ) goes to infinity.


    2. Relevant equations
    f(x) = 1/x
    f´(x)= -1/x^2

    Volume = [itex]\pi[/itex] [itex]^{\infty}_{1}[/itex][itex]\int[/itex] ((f(x))^2 )dx


    Area = 2[itex]\pi[/itex] [itex]^{\infty}_{1}[/itex][itex]\int[/itex] | f(x) | * [itex]\sqrt{}[/itex](1+(f´(x))^2) dx
    3. The attempt at a solution

    First page
    Second page

    I get the Volume done nicely, but the area? I know i could approximate the √(1+(1/x^4)) = √1 and it would solve easily, but what I'm doing wrong in my integral there? If we insert for example the s = 1, we get ln( 1-1 ) which we know ain't allowed.

    So I think my integral is totally off but can't figure out how.

    Sincerely yours,
    Siune
     
  2. jcsd
  3. Jan 29, 2012 #2
    You were on the right track, but it gets to be a messy integral. I would just use the comparison test for integrals.

    I don't see where the (-1/2) on the second page, first line went. Your integral seems fine except that you're missing a 1/2 on all your terms. The next step would be to sub back in your substitutions and clean up your answer.

    Edit: I also don't understand your limits of integration. You're still integrating from 1 to infinity. If I read your writing correctly, you have them set up from sqrt(2) to 1 on the second page, why?
     
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