delirija7z

2022-07-01

How do you find the average value of $f\left(x\right)=\mathrm{cos}x$ as x varies betwen $\left[0,\frac{\pi }{2}\right]$?

Charlee Gentry

Expert

The average value of the function f(x) on the interval [a,b] can be evaluated through the following the following expression:
average value $=\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)dx$
Here, this gives us an average value of:
$\frac{1}{\frac{\pi }{2}-0}{\int }_{0}^{\frac{\pi }{2}}\mathrm{cos}\left(x\right)dx$
Integrating $\mathrm{cos}\left(x\right)$ gives us $\mathrm{sin}\left(x\right):$
$=\frac{1}{\frac{\pi }{2}}\left[\mathrm{sin}\left(x\right){\right]}_{0}^{\frac{\pi }{2}}$
$=\frac{2}{\pi }\left[\mathrm{sin}\left(\frac{\pi }{2}\right)-\mathrm{sin}\left(0\right)\right]$
$=\frac{2}{\pi }\left[1-0\right]$
$=\frac{2}{\pi }$

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