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

[natemac42]

Homework Statement

The function ln(x) is rotated around the x-axis on the interval [1,4].

Homework Equations

Find the volume of the figure using integration.

The Attempt at a Solution

$$\pi$$ $$\int _{1}^{4} (.75) ln(x)^{2} dx$$

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

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:

$$dV = \pi*r^2 dx$$

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

$$dV = \pi(ln|x|)^2 dx$$

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