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Revolution about the y-axis

  1. Jan 17, 2009 #1
    1. The problem statement, all variables and given/known data
    Let R be the region bounded by the graph y=(1/x)ln(x), the x-axis, and the line x=e.
    Find the volume of the solid formed by revolving the region R about the y-axis.

    The interval should be (on the x-axis) from 1 to e
    and from the y-axis, it should be from 1 to (1/e)

    the area of the region is 1/2

    2. Relevant equations
    [tex]V=\pi\int(top^2-bottom^2) over an interval [/tex]
    (at least, that is what I use when rotating an area over the x-axis, I've never had the y-axis before)

    3. The attempt at a solution
    I was able to get the area quite easily, but when reading on how to find the volume, it said to write the equations in terms of y, so Iwas able to get f(y)=e REALLY easily, but I am having problems getting the other equation in terms of y, and I'm not even sure one can do so! If this is the case, how should I approach solving this problem? Thanks for any replies.
    Last edited: Jan 17, 2009
  2. jcsd
  3. Jan 17, 2009 #2


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    It's a LOT easier to do this problem using the method of shells rather than the disk or washer method. Integrate the area of a cylindrical shell from 0 to e.
  4. Jan 17, 2009 #3
    alright, I understand a little better now, thank you, but it still seems that I have to take that nasty function with a natural log, y=(1/x)ln(x), and write that in terms of y, which I am having problems doing. Is that possible or am I still headed downthe wrong path. :confused:
  5. Jan 17, 2009 #4


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    That's the point to using cylinders, you don't have to solve for x in terms of y. The height of the cylinder is (1/x)*ln(x). What's the radius of cylinder?
  6. Jan 17, 2009 #5
    Wouldn't that be dy? hmmmm.... I think I may be able to get it with a little more brain power! Thank you very much for your help.
  7. Jan 17, 2009 #6


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    I'll give you a blunt hint. The radius is x.
  8. Jan 17, 2009 #7
    yeeeeeup. After a few more searches on google and some rought drawings, I figured it out!
    My formula should be :
    [tex]V=2\pi\int(x((1/x)ln(x))dx) from 1 to e[/tex]
    which becomes...
    [tex]V=2\pi\int(ln(x))dx from 1 to e[/tex]

    So! Thank you very much again for your help! It is greatly appreciated.
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