How Do I Find the Volume of a Solid Revolved Around the y-Axis?

[Swerting]

Homework Statement

Let R be the region bounded by the graph y=(1/x)ln(x), the x-axis, and the line x=e.
Find the volume of the solid formed by revolving the region R about the y-axis.

The interval should be (on the x-axis) from 1 to e
and from the y-axis, it should be from 1 to (1/e)

the area of the region is 1/2

Homework Equations

$$V=\pi\int(top^2-bottom^2) over an interval$$
(at least, that is what I use when rotating an area over the x-axis, I've never had the y-axis before)

The Attempt at a Solution

I was able to get the area quite easily, but when reading on how to find the volume, it said to write the equations in terms of y, so Iwas able to get f(y)=e REALLY easily, but I am having problems getting the other equation in terms of y, and I'm not even sure one can do so! If this is the case, how should I approach solving this problem? Thanks for any replies.

 

[Dick]

It's a LOT easier to do this problem using the method of shells rather than the disk or washer method. Integrate the area of a cylindrical shell from 0 to e.

 

[Swerting]

alright, I understand a little better now, thank you, but it still seems that I have to take that nasty function with a natural log, y=(1/x)ln(x), and write that in terms of y, which I am having problems doing. Is that possible or am I still headed downthe wrong path.

 

[Dick]

That's the point to using cylinders, you don't have to solve for x in terms of y. The height of the cylinder is (1/x)*ln(x). What's the radius of cylinder?

 

[Swerting]

That's the point to using cylinders, you don't have to solve for x in terms of y. The height of the cylinder is (1/x)*ln(x). What's the radius of cylinder?

Wouldn't that be dy? hmmmm... I think I may be able to get it with a little more brain power! Thank you very much for your help.

 

[Dick]

Wouldn't that be dy? hmmmm... I think I may be able to get it with a little more brain power! Thank you very much for your help.

I'll give you a blunt hint. The radius is x.

 

[Swerting]

I'll give you a blunt hint. The radius is x.

yeeeeeup. After a few more searches on google and some rought drawings, I figured it out!
My formula should be :
$$V=2\pi\int(x((1/x)ln(x))dx) from 1 to e$$
which becomes...
$$V=2\pi\int(ln(x))dx from 1 to e$$

So! Thank you very much again for your help! It is greatly appreciated.