6.5 Average Value of a Function/1

Find the average value of the function on the given interval.

$f(x)=4x-x^{2},\quad (0,4)$

${\begin{aligned}f_{avg}&={\frac {1}{4-0}}\int _{0}^{4}(4x-x^{2})dx\\[2ex]&={\frac {1}{4}}\int _{0}^{4}(4x-x^{2})dx\\[2ex]&={\frac {1}{4}}\left[{\bigg (}4x-{\frac {x^{3}}{3}}{\bigg )}\right]_{0}^{4}\\[2ex]&={\frac {1}{4}}{\bigg (}8-{\frac {8}{3}}{\bigg )}-{\bigg (}-8+{\frac {8}{3}}{\bigg )}\\[2ex]&={\frac {1}{4}}\left[{\bigg (}{\frac {32}{4}}{\bigg )}\right]\\[2ex]&={\frac {8}{3}}\end{aligned}}$