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Compute the average value of the function on the interval

Compute the average value of

    \[ f(x) = \sin (2x), \qquad 0 \leq x \leq \frac{\pi}{2}. \]


We compute the average A(f),

    \[ A(f) = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \sin (2x) \, dx = \frac{1}{\pi} \int_0^{\pi} \sin x \, dx = \frac{1}{\pi}(1 - \cos \pi) = \frac{2}{\pi}.\]

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