ParametricPlot3D[{-Sin[s] (2 + Sqrt[1 - (t - 2)^2]), t, 
        Cos[s] (2 + Sqrt[1 - (t - 2)^2])}, {t, 1, 3}, {s, 0, 
        2  Pi}, PlotPoints -> 70]

f[t_] = 2 + Sqrt[1 - (t - 2)^2]

1 + f'[t]^2

Simplify[1 + f'[t]^2]

2 Pi Integrate[
        f[t]*Sqrt[1 + ((((-2 + t))^2)/(1 - (((-2 + t))^2)))], {t, 
          1, 3}]

g[t_] = 2 - Sqrt[1 - (t - 2)^2]

Simplify[1 + g'[t]^2]

2 Pi Integrate[
        g[t]*Sqrt[1 + ((((-2 + t))^2)/(1 - (((-2 + t))^2)))], {t, 
          1, 3}]