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Compute the average value of the function on the interval

Compute the average value of

    \[ f(x) = \cos^2 x, \qquad 0 \leq x \leq \pi. \]


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

    \begin{align*}  A(f) = \frac{1}{\pi} \int_0^{\pi} cos^2 x \, dx &= \frac{1}{2 \pi} \int_0^{\pi} (1+\cos 2x) \, dx \\  &= \frac{1}{2 \pi} (\pi + 0) \\  &= \frac{1}{2}. \end{align*}

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