Answer
$$S = \frac{8}{3}\pi \left[ {10\sqrt {10} - 5\sqrt 5 } \right]$$
Work Step by Step
$$\eqalign{
& y = 2\sqrt x ,{\text{ }}4 \leqslant x \leqslant 9 \cr
& {\text{Differentiate}} \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {2\sqrt x } \right] \cr
& \frac{{dy}}{{dx}} = 2\left( {\frac{1}{{2\sqrt x }}} \right) \cr
& \frac{{dy}}{{dx}} = \frac{1}{{\sqrt x }} \cr
& {\text{Formula for surface area}} \cr
& S = 2\pi \int_a^b {r\left( x \right)\sqrt {1 + {{\left[ {f'\left( x \right)} \right]}^2}} } dx,{\text{ }}r\left( x \right) = f\left( x \right) \cr
& {\text{Then,}} \cr
& S = 2\pi \int_4^9 {2\sqrt x \sqrt {1 + {{\left[ {\frac{1}{{\sqrt x }}} \right]}^2}} } dx \cr
& S = 2\pi \int_4^9 {2\sqrt x \sqrt {1 + \frac{1}{x}} } dx \cr
& S = 2\pi \int_4^9 {2\sqrt x \sqrt {\frac{{x + 1}}{x}} } dx \cr
& S = 2\pi \int_4^9 {\frac{{2\sqrt x }}{{\sqrt x }}\sqrt {x + 1} } dx \cr
& S = 4\pi \int_4^9 {\sqrt {x + 1} } dx \cr
& {\text{Integrate}} \cr
& S = 4\pi \left[ {\frac{{{{\left( {x + 1} \right)}^{3/2}}}}{{3/2}}} \right]_4^9 \cr
& S = \frac{8}{3}\pi \left[ {{{\left( {x + 1} \right)}^{3/2}}} \right]_4^9 \cr
& S = \frac{8}{3}\pi \left[ {{{\left( {9 + 1} \right)}^{3/2}} - {{\left( {4 + 1} \right)}^{3/2}}} \right] \cr
& S = \frac{8}{3}\pi \left[ {10\sqrt {10} - 5\sqrt 5 } \right] \cr
& S \approx 171.2581 \cr} $$
