Definition:Average Value of Function

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.