Compute the average value of the function on the interval

by RoRi

August 16, 2015

Compute the average value of

$f(x) = \sin x \cos x, \qquad 0 \leq x \leq \frac{\pi}{4}.$

Recalling that $\frac{1}{2} \sin (2x) = \sin x \cos x$ , we compute the average $A(f)$ ,

$\begin{align*} \frac{4}{\pi} \int_0^{\frac{\pi}{4}} \sin x \cos x \, dx &= \frac{2}{\pi} \int_0^{\frac{\pi}{4}} \sin (2x) \, dx \\ &= \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \sin x \, dx \\ &= \frac{1}{\pi}\left( 1 - \cos \frac{\pi}{2} \right) \\ &= \frac{1}{\pi}. \end{align*}$

Stumbling Robot

Compute the average value of the function on the interval

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