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