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

  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. rock.freak667

    rock.freak667 6,221
    Homework Helper

    try letting

    u=lnx and then go from there
  4. 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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