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Compute the average value of the function on the interval

Compute the average value of

    \[ f(x) = x^{1/3}, \qquad 1 \leq x \leq 8. \]


We compute the average A(f),

    \[ A(f) = \frac{1}{8-1} \int_1^8 x^{1/3} \, dx = \frac{1}{7} \cdot \frac{3}{4} \left( x^{4/3} \biggr \rvert_1^8 \right) = \frac{3}{28} (16 - 1) = \frac{45}{28}. \]

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