Answer
\[L = \frac{1}{27}\,\,\,\left[ {{{22}^{\frac{3}{2}}} - 8} \right]\]
Work Step by Step
\[\begin{gathered}
\,find\,\,the\,length\,\,of\,\,the\,\,curve \hfill \\
\hfill \\
Let\,\,y = {x^{\frac{3}{2}}} + 8\,\,on\,\,the\,\,interval\,\,\,\,\left[ {0,2} \right] \hfill \\
\hfill \\
\,\,the\,\,length\,of\,the\,\,curve\,\,y = \,\left( x \right)\,\,is \hfill \\
\hfill \\
{\text{ }}\,L = \int_a^b {\sqrt {1 + {{y'}^2}} } \,dx \hfill \\
\hfill \\
y = {x^{\frac{3}{2}}} + 8\,\,\,\,\,\,then\,\,differentiating,\,\,\,y' = \frac{3}{2}\sqrt x \hfill \\
\hfill \\
L = \int_0^2 {\sqrt {1 + \,{{\left( {\frac{3}{2}\sqrt x } \right)}^2}} dx} \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
= \int_0^2 {\sqrt {4 + 9x} } \,dx \hfill \\
\hfill \\
use\,\,the\,\,formula \hfill \\
\hfill \\
\int_{}^{} {\sqrt {ax + b} \,dx} \,\, = \frac{2}{{3a}}\,{\left( {ax + b} \right)^{\frac{3}{2}}} + C \hfill \\
\hfill \\
then \hfill \\
\hfill \\
L = \int_0^2 {\sqrt {4 + 9x} } dx = \frac{1}{2}\,\,\left[ {\frac{2}{{27}}\,{{\left( {4 + 9x} \right)}^{\frac{3}{2}}}} \right]_0^2 \hfill \\
\hfill \\
evaluate\,\,the\,\,{\text{limits}}\,\,and\,\,simplify \hfill \\
\hfill \\
L = \frac{1}{27}\,\,\,\left[ {{{22}^{\frac{3}{2}}} - 8} \right] \hfill \\
\end{gathered} \]
