average value of a function

The height of the flat rectangle whose area over an interval matches the actual area under 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 Mean Value Theorem for Integrals guarantees a continuous $f$ actually attains this average value at some point $c \in [a,b]$ .

Examples

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