No I am not. including it. The distance from the rotational axis to the nearest boundary on the region is 1 unit (the distance from y=1 to y=2) at every level x. Call this the inner radius r(x).
The distance from the rotational axis to the outer boundary of the region is the y-coordinate of your exponential function minus 1, due the the rotational axis being y=1, one unit above the horizontal axis. Call this R(x).
The washer method is a difference between two disks:
INTEGRAL (Pi R^2) - INTEGRAL (Pi r^2)
= Pi INTEGRAL (R^2 - r^2).
I have posted handout of the washer and disk methods on
my website.
A direct link of this document:
ap calculus disk washer methods - Fullscreen
Good luck!!