$\displaystyle y=\ln{x} , 1\leq x\leq e$

Calculate the volume between y and the x-axis with:

a) integration with respect to x.

b)integration with respect to y.

I did a) without any problem, but I don't get how to do with b).

The correct answer are $\displaystyle 2\pi \int_{0}^{1} y(e-e^y) dy$

How should I think? I get that I should change to equation from $\displaystyle y= \ln{x} , 1\leq x\leq e$ to $\displaystyle x=e^y , 0\leq y\leq 1$ and that the but then I got no idea at all...