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Average Value

The average value of an integrable function f on a finite interval [a,b] with a<b is

f_(av)=1/(b-a)int_a^bf(x)dx.

More generally, if E has finite positive measure mu(E), the average of f over E is

<f>_E=1/(mu(E))int_Efdmu.

If f is a real-valued function that is continuous on [a,b], the mean-value theorem for integrals guarantees a point c in [a,b] such that f(c)=f_(av).

See also

Average, Integral, Mean

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References

Apostol, T. M. Calculus, Vol. 1, 2nd ed. New York: Wiley, 1967.

Cite this as:

Weisstein, Eric W. "Average Value." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AverageValue.html

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