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Compute the average value of the function on the interval

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*}

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