Answer
$\frac{{{9}^{7}}-1}{6}=\frac{2,391,484}{3}$
Work Step by Step
\[\begin{align}
& f\left( x \right)=7{{x}^{2}}{{\left( {{x}^{3}}+1 \right)}^{6}},\text{ over the interval }\left[ 0,2 \right] \\
& \text{The average value is:} \\
& {{f}_{avg}}=\frac{1}{b-a}\int_{a}^{b}{f\left( x \right)}dx \\
& {{f}_{avg}}=\frac{1}{2-0}\int_{0}^{2}{7{{x}^{2}}{{\left( {{x}^{3}}+1 \right)}^{6}}}dx \\
& {{f}_{avg}}=\frac{7}{6}\int_{0}^{2}{3{{x}^{2}}{{\left( {{x}^{3}}+1 \right)}^{6}}}dx \\
& {{f}_{avg}}=\frac{7}{6}\left[ \frac{{{\left( {{x}^{3}}+1 \right)}^{7}}}{7} \right]_{0}^{2} \\
& {{f}_{avg}}=\frac{1}{6}\left[ {{\left( {{2}^{3}}+1 \right)}^{7}}-{{\left( {{0}^{3}}+1 \right)}^{7}} \right] \\
& {{f}_{avg}}=\frac{1}{6}\left[ {{\left( 9 \right)}^{7}}-{{\left( 1 \right)}^{7}} \right] \\
& {{f}_{avg}}=\frac{{{9}^{7}}-1}{6}=\frac{2,391,484}{3} \\
\end{align}\]
