How to visualise the curve $y=\ln x$ rotating around its $y$-axis 
Let the area limited by the curve $y=\ln x$, the line $x=e$ and the $x$-axis rotate around the $y$-axis. Decide the volume of the resulting rotational body.

First thing, I drew the graph:

But then I got stuck on how to imagine/visualise the curve rotating around its $y$-axis. How should I think when visualising the function $lnx$ rotating around its $y$-axis? 
EDIT: 
In the comment section it was written that "if you rotate the rectangle $[0,e]×[0,1]$, it makes a cylinder which volume is easy to compute", but I am not very familiar with the notation for $[0,e]×[0,1]$. 

Here I (the OP) will add my solution to the problem.
Think of it like follows. If the integral $\int_0^1 dy \;\pi (e)^2$ represents the area of 

And the integral $\int_0^1 dy \;\pi (e^y)^2$ represents the area of 

Then the area of 

is computed like this: 
$$\int_0^1 dy \;\pi (e)^2-\int_0^1 dy \;\pi (e^y)^2=\int_0^1 dy \;\pi(e^2-e^{2y})$$
And this integral gives you the answer $\frac{\pi (e^2+1)}{2} \text{v.u.}$
