Answer
$$\overline f = 0$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = {x^3}{\text{ on the interval }}\left[ { - 1,1} \right] \cr
& {\text{Find the average value using }}\overline f = \frac{1}{{b - a}}\int_a^b {f\left( x \right)} dx \cr
& \overline f = \frac{1}{{1 - \left( { - 1} \right)}}\int_{ - 1}^1 {{x^3}} dx \cr
& \overline f = \frac{1}{2}\int_{ - 1}^1 {{x^3}} dx \cr
& {\text{Integrate}} \cr
& \overline f = \frac{1}{2}\left[ {\frac{1}{4}{x^4}} \right]_{ - 1}^1 \cr
& \overline f = \frac{1}{8}\left[ {{{\left( 1 \right)}^4} - {{\left( { - 1} \right)}^4}} \right] \cr
& \overline f = 0 \cr
& \cr
& {\text{Graph}} \cr} $$
