Answer
$$\left. a \right)L = \int_{ - 1}^1 {\sqrt {1 + {x^4}} } dx,\,\,\,\,\left. b \right)\,\,L \approx 2.18$$
Work Step by Step
$$\eqalign{
& y = \frac{{{x^3}}}{3}{\text{ on the interval }}\left[ { - 1,1} \right] \cr
& {\text{Definition of Arc Length for }}y = f\left( x \right): \cr
& {\text{Let }}f{\text{ have a continuous first derivative on the interval }}\left[ {a,b} \right]{\text{ }} \cr
& {\text{The length of the curve }} \cr
& {\text{from }}\left( {a,f\left( a \right)} \right){\text{ to }}\left( {b,f\left( b \right)} \right){\text{ is }}L = \int_a^b {\sqrt {1 + f'{{\left( x \right)}^2}} } dx \cr
& {\text{Notice that }}y = f\left( x \right) = \ln x{\text{ and }} \cr
& \left[ { - 1,1} \right] \to a = - 1{\text{ and }}b = 1.{\text{ Then}} \cr
& \cr
& \left. a \right)\,\,f'\left( x \right) = \frac{d}{{dx}}\left[ {\frac{{{x^3}}}{3}} \right] \cr
& f'\left( x \right) = \frac{{3{x^2}}}{3} \cr
& f'\left( x \right) = {x^2} \cr
& {\text{Using the arc length formula}}{\text{, we have}} \cr
& L = \int_{ - 1}^1 {\sqrt {1 + {{\left( {{x^2}} \right)}^2}} } dx \cr
& L = \int_{ - 1}^1 {\sqrt {1 + {x^4}} } dx \cr
& \cr
& \left. b \right){\text{ Use technology to evaluate or approximate the integral}} \cr
& L \approx 2.18 \cr} $$