Answer
$$S = 35\sqrt 2 \pi $$
Work Step by Step
$$\eqalign{
& y = 7x,\,\,\,\,0 \leqslant x \leqslant 1 \cr
& {\text{Calculate }}\frac{{dy}}{{dx}} \cr
& \frac{{dy}}{{dx}} = 7 \cr
& {\text{Use }}S = \int_a^b {2\pi y\sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} } dx \cr
& S = \int_0^1 {2\pi \left( {7x} \right)\sqrt {1 + {{\left( 7 \right)}^2}} } dx \cr
& S = \int_0^1 {14\pi x\sqrt {50} } dx \cr
& S = 70\sqrt 2 \pi \int_0^1 x dx \cr
& {\text{Integrate}} \cr
& S = 70\sqrt 2 \pi \left[ {\frac{1}{2}{x^2}} \right]_0^1 \cr
& S = 35\sqrt 2 \pi \left[ {{x^2}} \right]_0^1 \cr
& S = 35\sqrt 2 \pi \cr} $$
