the mean value of a function

The mean (average) value of a function f(x) over the interval 0 to a is the average of n values of the function uniformly distributed over the interval, as n increases without limit, i.e.

    \[ \M_{0}^{a}f(x)=\lim_{n\to\infty}\frac{f(\delta)+f(2\delta)+\dotsb+f(n\delta)}{n}  \]

where \delta =a/n, the nth part of a.

e.g. If a car starts moving forward, and accelerates (with constant acceleration) until it’s travelling 100km/h, the mean (average) speed of the journey is 50km/h.

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