It certainly is "doable". The region in the xy-plane is a "quadrilateral with one curved side, vertices at (1, 0), (2, 0), and (1, e). There would be a problem with doing it as "washers" since there would be a change of formula at (1, e).
If you "slice" it parallel to the y-axis, rotating around the y-axis, each "slice" would be a cylinder, of thickness dx, and height the distance from y= 0 to which is .
Since this is rotated around the x-axis, the radius of each cylinder is x and the area is . The volume of each thin cylinder is and the entire volume is given by the integral