The integral mean

Before stating the theorem, we need to define what is meant by the average value of a function over an interval.

Definition — Integral mean

The integral mean of $f$ on $[a,b]$ is: $\bar f_{ab} = \frac{1}{b-a}\int_a^b f(x)\,dx.$ Geometrically it is the height of the rectangle with base $(b-a)$ that has the same area as the region under the graph.

The area under the graph of $f$ (in light blue) equals the area of the rectangle of height $\bar f$ (dashed red line) and base $(b-a)$.

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Topics: Calculus theorems
Concepts: Definite integral · Integral mean
Skills: Integrating