landdenaw

2022-06-22

How do you find the average value of $\mathrm{sin}x$ as x varies between $\left[0,\pi \right]$?

Jaida Sanders

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

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