# Definition:Average Value of Function

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## Definition

Let $f$ be an integrable function on some closed interval $\closedint a b$.

The **average value of $f$** (or **mean value of $f$**) on $\closedint a b$ is defined as:

- $\ds \frac 1 {b - a} \int_a^b \map f x \rd x$

## Also see

- Mean Value Theorem for Integrals which proves that $f$ attains this value on $\closedint a b$, provided additionally that $f$ is continuous on $\closedint a b$.

## 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.): $\S 4.4$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**mean value**