Answer
$$S \approx 22.943$$
Work Step by Step
$$\eqalign{
& y = \ln x,{\text{ }}1 \leqslant x \leqslant e \cr
& {\text{Differentiate}} \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {\ln x} \right] \cr
& \frac{{dy}}{{dx}} = \frac{1}{x} \cr
& {\text{Therefore,}} \cr
& S = 2\pi \int_a^b {r\left( x \right)\sqrt {1 + {{\left[ {f'\left( x \right)} \right]}^2}} } dx,{\text{ }}r\left( x \right) = x \cr
& {\text{Then,}} \cr
& S = 2\pi \int_1^e {x\sqrt {1 + {{\left( {\frac{1}{x}} \right)}^2}} } dx \cr
& S = 2\pi \int_1^e {\frac{x}{x}\sqrt {{x^2} + 1} } dx \cr
& S = 2\pi \int_1^e {\sqrt {{x^2} + 1} } dx \cr
& {\text{Integrate using a CAS or graphing utility}} \cr
& S \approx 22.943 \cr} $$
