Answer
$$L = \frac{{85\sqrt {85} - 8}}{{243}}$$
Work Step by Step
$$\eqalign{
& y = 3{x^{3/2}} - 1{\text{ from }}x = 0{\text{ to }}x = 1 \cr
& {\text{Calculate }}\frac{{dy}}{{dx}} \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {3{x^{3/2}} - 1} \right] \cr
& \frac{{dy}}{{dx}} = \frac{9}{2}{x^{1/2}} \cr
& {\text{Use }}L = \int_a^b {\sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} dx} \cr
& L = \int_0^1 {\sqrt {1 + {{\left( {\frac{9}{2}{x^{1/2}}} \right)}^2}} dx} \cr
& L = \int_0^1 {\sqrt {1 + \frac{{81}}{4}x} dx} \cr
& L = \frac{1}{2}\int_0^1 {\sqrt {4 + 81x} dx} \cr
& {\text{Integrating}} \cr
& L = \frac{1}{{2\left( {81} \right)}}\left[ {\frac{{2{{\left( {4 + 81x} \right)}^{3/2}}}}{3}} \right]_0^1 \cr
& L = \frac{1}{{243}}\left[ {{{\left( {4 + 81} \right)}^{3/2}} - {{\left( {4 + 0} \right)}^{3/2}}} \right] \cr
& L = \frac{{{{85}^{3/2}} - 8}}{{243}} \cr
& L = \frac{{85\sqrt {85} - 8}}{{243}} \cr} $$
