Answer
$$S = 40\pi \sqrt {82} $$
Work Step by Step
$$\eqalign{
& x = 9y + 1,\,\,\,\,0 \leqslant y \leqslant 2 \cr
& {\text{Calculate }}\frac{{dx}}{{dy}} \cr
& \frac{{dx}}{{dy}} = 9 \cr
& {\text{Use }}S = \int_a^b {2\pi x\sqrt {1 + {{\left( {\frac{{dx}}{{dy}}} \right)}^2}} } dy \cr
& S = \int_0^2 {2\pi \left( {9y + 1} \right)\sqrt {1 + {{\left( 9 \right)}^2}} } dy \cr
& S = 2\pi \sqrt {82} \int_0^2 {\left( {9y + 1} \right)} dy \cr
& {\text{Integrate}} \cr
& S = 2\pi \sqrt {82} \left[ {\frac{{9{y^2}}}{2} + y} \right]_0^2 \cr
& S = 2\pi \sqrt {82} \left[ {\frac{{9{{\left( 2 \right)}^2}}}{2} + 2} \right] \cr
& S = 40\pi \sqrt {82} \cr} $$
