6.5 Average Value of a Function/1

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

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

${\begin{aligned}f_{avg}&={\frac {1}{4}}\int _{0}^{4}(4x-x^{2})dx\\[2ex]&={\frac {1}{4}}\left[{\bigg (}4x-{\frac {x^{3}}{3}}{\bigg )}\right]={\frac {1}{4}}{\bigg (}8-{\frac {8}{3}}{\bigg )}\\[2ex]&={\frac {1}{4}}[{\frac {32}{4}}]\\[2ex]&=38{\frac {1}{3}}\end{aligned}}$