Answer
$\frac{13}{6}$
Work Step by Step
\[\begin{align}
& f\left( x \right)=\sqrt{x+1},\text{ over the interval }\left[ 0,8 \right] \\
& \text{The average value is:} \\
& {{f}_{avg}}=\frac{1}{b-a}\int_{a}^{b}{f\left( x \right)}dx \\
& {{f}_{avg}}=\frac{1}{8-0}\int_{0}^{8}{\sqrt{x+1}}dx \\
& {{f}_{avg}}=\frac{1}{8}\left[ \frac{2}{3}{{\left( x+1 \right)}^{3/2}} \right]_{0}^{8} \\
& {{f}_{avg}}=\frac{1}{12}\left[ {{\left( 8+1 \right)}^{3/2}}-{{\left( 0+1 \right)}^{3/2}} \right] \\
& {{f}_{avg}}=\frac{1}{12}\left[ 27-1 \right] \\
& {{f}_{avg}}=\frac{13}{6} \\
\end{align}\]
