Ramiro Wise

2022-11-28

Find the average value fave of the function f on the given interval.
$f\left(t\right)=e\mathrm{sin}\left(t\right),\mathrm{cos}\left(t\right),\left[0,\frac{\pi }{2}\right]$

rezvanifanFdT

Expert

The average value of a function over an interval $\left[a,b\right]$ is given by
$\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)dx$
We have
$\frac{1}{\frac{\pi }{2}-0}{\int }_{0}^{\pi /2}{e}^{\mathrm{sin}t}\mathrm{cos}tdt=\frac{2}{\pi }{\int }_{0}^{\pi /2}{e}^{\mathrm{sin}t}d\left(\mathrm{sin}t\right)\phantom{\rule{0ex}{0ex}}=\frac{2}{\pi }{\int }_{\mathrm{sin}0}^{\mathrm{sin}\left(\pi /2\right)}{e}^{u}du=\frac{2}{\pi }{e}^{u}{|}_{\mathrm{sin}0}^{\mathrm{sin}\left(\pi /2\right)}\phantom{\rule{0ex}{0ex}}=\frac{2\left({e}^{1}-{e}^{0}\right)}{\pi }=\frac{2\left(e-1\right)}{\pi }$

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