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

natemac42

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

rock.freak667

try letting

u=lnx and then go from there

Feldoh

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?