Ln(x) rotated around the x-axis [1,4] Find Volume

  1. Jan 27, 2008 #1
    1. The problem statement, all variables and given/known data
    The function ln(x) is rotated around the x-axis on the interval [1,4].


    2. Relevant equations
    Find the volume of the figure using integration.


    3. The attempt at a solution
    [tex]\pi[/tex] [tex]\int _{1}^{4} (.75) ln(x)^{2} dx[/tex]

    = [tex]\pi[/tex] [tex]\int _{1}^{4} [3(ln(x))^{2}]/4 [/tex]

    sorry I'm bad at typing these things in


    anyway solving that I got 6.1187 units^3 and I don't think it's the correct answer, but I'm not sure.

    I approximated the volume using cylinders and got 10.518 for circumscribed and 5.989 for inscribed.

    Thanks in advance
     
  2. jcsd
  3. Jan 27, 2008 #2

    rock.freak667

    User Avatar
    Homework Helper

    try letting

    u=lnx and then go from there
     
  4. Jan 27, 2008 #3
    We can solve this by breaking the object created by breaking it into small parts. Since we're rotating the function around an axis we'll get a cylindrical-ish object. So we can find the area of a slab of the object by multiplying the area by an infinitely small width (dx).

    So an infinitely small piece of the solid is the area by the width:

    [tex]dV = \pi*r^2 dx[/tex]

    One problem though. The radius of these slabs is constantly changing according to ln(x)

    [tex]dV = \pi(ln|x|)^2 dx[/tex]

    [tex]V = \int _{1}^{4} \pi(ln|x|)^2 dx[/tex] is what I'm getting?
     
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