Average value of a function

$\dfrac{1}{b-a}\int_a^b f(x)\,dx$ — the height of the rectangle with the same area as the curve.

The average value of $f$ on $[a,b]$ is $f_{\text{avg}}=\dfrac{1}{b-a}\int_a^b f(x)\,dx$ — the constant height a rectangle over $[a,b]$ would need to match the curve's net signed area exactly. It generalizes the everyday average (sum divided by count) to a continuum of infinitely many values.

Examples

The average value of $f(x)=x^2$ on $[0,3]$ is $\dfrac{1}{3}\int_0^3 x^2\,dx=\dfrac{1}{3}\cdot9=3$ .