Average Value of Function

• Intuitively, this is $$\lim_{ n \to \infty } \frac{\sum_{i=1}^{n}f(x)}{n}$$
• Multiplying the top and bottom by $(b-a)$, we get $$\frac{\lim_{ n \to \infty } \sum_{i=0}^{n}f(x) \frac{b-a}{n}}{b-a}$$
• The top is a Riemann Sum, so we can convert it to an Integral $$
  \overline{f}=\frac{1}{b-a}\int_{a}^{b} f(x) , dx

\begin{align}
m & \leq f(x)\leq M \
m(b-a) & \leq \int_{a}^{b} f(x) , dx \leq M(b-a) \
m & \leq \frac{1}{b-a}\int_{a}^{b} f(x) , dx\leq M
\end