Answer
$$\overline f = \frac{7}{3}$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = {x^2} + 1{\text{ on the interval }}\left[ { - 2,2} \right] \cr
& {\text{Find the average value using }}\overline f = \frac{1}{{b - a}}\int_a^b {f\left( x \right)} dx \cr
& \overline f = \frac{1}{{2 - \left( { - 2} \right)}}\int_{ - 2}^2 {\left( {{x^2} + 1} \right)} dx \cr
& \overline f = \frac{1}{4}\left[ {\frac{1}{3}{x^3} + x} \right]_{ - 2}^2 \cr
& {\text{Integrate}} \cr
& \overline f = \frac{1}{4}\left[ {\frac{1}{3}{{\left( 2 \right)}^3} + \left( 2 \right)} \right] - \frac{1}{4}\left[ {\frac{1}{3}{{\left( { - 2} \right)}^3} + \left( { - 2} \right)} \right] \cr
& \overline f = \frac{1}{4}\left( {\frac{{14}}{3}} \right) - \frac{1}{4}\left( { - \frac{{14}}{3}} \right) \cr
& \overline f = \frac{7}{3} \cr
& \cr
& {\text{Graph}} \cr} $$
