Set up, but do not evaluate, a definite integral representing the volume of the

Question

solid formed by revolving the region of the plane bounded by the curves y = e
x
, x = 0 and
y = e about the line x = 1 using the disk/washer method.

Answer 1

you have washers of thickness dy, so
v = ∫[1,e] π(R^2 - r^2) dy
where R = 1 and r = 1-x = 1-lny
v = ∫[1,e] π(1^2 - (1 - lny)^2) dy

just as a check, using shells of thickness dx,
v = ∫[0,1] 2πrh dx
where r = 1-x and h = e-y = e-e^x
v = ∫[0,1] 2π(1-x)(e-e^x) dx