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