Calc1.78

Find the average value of the function $f(x)=e^{{2x}}$ on the interval $[0,4]$.
${\frac {1}{b-a}}\int _{{a}}^{{b}}f(x)\,dx\,$
${\frac {1}{4-0}}\int _{{0}}^{{4}}e^{{2x}}\,dx={\frac {1}{4}}\left[{\frac {1}{2}}e^{{2x}}\right]_{{0}}^{{4}}={\frac {1}{8}}\left[e^{8}-1\right]\,$
Thus, the average value of $f(x)$ on the interval $[0,4]$ is ${\frac {1}{8}}\left[e^{8}-1\right]$. Though the function increases rapidly and thus is different on every interval, its average value per 1 unit, on this interval, is about 372.49.