Answer
$$S = \frac{{\left( {30\sqrt {30} - 10\sqrt {10} } \right)\pi }}{3}$$
Work Step by Step
$$\eqalign{
& y = \sqrt {4x + 6} {\text{ on }}\left[ {0,5} \right] \cr
& {\text{use the Definition of Area of a Surface of Revolution }}\left( {{\text{see page 454}}} \right) \cr
& {\text{The area of the surface generated when the graph of }}f{\text{ on the interval }}\left[ {a,b} \right] \cr
& {\text{is revolved about the }}x - {\text{axis}} \cr
& S = \int_a^b {2\pi f\left( x \right)\sqrt {1 + f'{{\left( x \right)}^2}} } dx \cr
& {\text{then}} \cr
& y = f\left( x \right) = 8\sqrt x {\text{ and }}\left[ {0,5} \right] \to a = 0{\text{ and }}b = 5.{\text{ then}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {\sqrt {4x + 6} } \right] \cr
& f'\left( x \right) = \frac{4}{{2\sqrt {4x + 6} }} = \frac{2}{{\sqrt {4x + 6} }} \cr
& {\text{substituting the values into the formula }}S = \int_a^b {2\pi f\left( x \right)\sqrt {1 + f'{{\left( x \right)}^2}} } dx \cr
& S = \int_0^5 {2\pi \left( {\sqrt {4x + 6} } \right)\sqrt {1 + {{\left( {\frac{2}{{\sqrt {4x + 6} }}} \right)}^2}} } dx \cr
& S = 2\pi \int_0^5 {\left( {\sqrt {4x + 6} } \right)\sqrt {1 + \frac{4}{{4x + 6}}} } dx \cr
& S = 2\pi \int_0^5 {\left( {\sqrt {4x + 6} } \right)\sqrt {\frac{{4x + 6 + 4}}{{4x + 6}}} } dx \cr
& S = 2\pi \int_0^5 {\left( {\sqrt {4x + 6} } \right)\frac{{\sqrt {4x + 10} }}{{\sqrt {4x + 6} }}} dx \cr
& S = 2\pi \int_0^5 {\sqrt {4x + 10} } dx \cr
& S = \frac{{2\pi }}{4}\int_0^5 {\sqrt {4x + 10} \left( 4 \right)} dx \cr
& {\text{integrate and evaluate}} \cr
& S = \frac{\pi }{2}\left( {\frac{{{{\left( {4x + 10} \right)}^{3/2}}}}{{3/2}}} \right)_0^5 \cr
& S = \frac{\pi }{3}\left( {{{\left( {4x + 10} \right)}^{3/2}}} \right)_0^5 \cr
& S = \frac{\pi }{3}\left( {{{\left( {4\left( 5 \right) + 10} \right)}^{3/2}} - {{\left( {4\left( 0 \right) + 10} \right)}^{3/2}}} \right) \cr
& {\text{simplifying}} \cr
& S = \frac{\pi }{3}\left( {{{\left( {30} \right)}^{3/2}} - {{\left( {10} \right)}^{3/2}}} \right) \cr
& S = \frac{\pi }{3}\left( {30\sqrt {30} - 10\sqrt {10} } \right) \cr
& S = \frac{{\left( {30\sqrt {30} - 10\sqrt {10} } \right)\pi }}{3} \cr} $$
