\begin{frame} \frametitle{Average Value of a Function} If we let $n$ go to infinity: \begin{talign} \lim_{n \to \infty} \frac{1}{b-a}\sum_{i = 1}^n f(x_i)\Delta x \end{talign} we get\pause \begin{talign} \frac{1}{b-a} \int_a^b f(x) \, dx \end{talign} \pause As a consequence, we have \begin{block}{} The average value $f_{\text{avg}}$of a function $f$ on an interval $[a,b]$ is: \begin{talign} f_{\text{avg}} = \frac{1}{b-a} \int_a^b f(x) \, dx \end{talign} Thus the average value of the function is the integral over the interval divided by the width of the interval. \end{block} \end{frame}