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

by natemac42
Tags: rotated, volume, xaxis
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natemac42
#1
Jan27-08, 06:48 PM
P: 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
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rock.freak667
#2
Jan27-08, 07:20 PM
HW Helper
P: 6,205
try letting

u=lnx and then go from there
Feldoh
#3
Jan27-08, 09:19 PM
P: 1,345
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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