I am attaching the region in a pdf file.

Use the washer method for this problem, or the shell method if you are familiar with both. Below I quickly discuss the shell method:

V = INTEGRAL ( distance traveled * height of region)

distance = 2 pi x since the radius from the rotational axis to a level x is simply the x coordinate

height of region = 1 - ln x

the bounds are from x = 1 to the intersection point between y=ln x and y=1, which is simply e = 2.71828...

So the volume is INTEGRAL (2pi x (1-ln x)) from 1 to e.

The washer method, briefly:

Inner radius: r(y) = 1

Outer radius R(y) = x = e^y

Bounds y = 0 to y =1

Formula: V = INTEGRAL (Pi R^2 - Pi r^2)

Good luck!!