# Definition:Average Value of Function

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## Definition

Let $f$ be an integrable function on some closed interval $\left[{a \,.\,.\, b}\right]$.

The **average value of $f$** (or **mean value of $f$**) on the interval is defined as:

- $\displaystyle \frac 1 {b-a}\int_a^b f \left({x}\right) \ \mathrm d x$

## Also see

- Mean Value Theorem for Integrals which proves that such a number exists.

## Note on Terminology

The word average is generally considered to be too vague for use in mathematics, as it could mean one of a number of *kinds* of average.

For serious mathematics it is considered preferable to use the term **mean value** rather than **average value**.

However, this is a significant elementary concept which has applications across a wide range of applied mathematics and soft-science subjects, and the popular terminology in such circumstances takes precedence.

## Sources

- 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards:
*Calculus*(8th ed.): $\S 4.4$