\begin{frame} \frametitle{Symmetry} Symmetry can sometimes help to simplify integrals! \pause \begin{block}{} Suppose $f$ is continuous on $[-a,a]$: \pause \begin{itemize} \item If $f$ is even [$f(-x) = f(x)$], then \begin{talign} \int_{-a}^a f(x)\,dx = \mpause[1]{2\int_0^a f(x)\,dx} \end{talign} \pause\pause\vspace{-1ex} \item If $f$ is odd [$f(-x) = -f(x)$], then \begin{talign} \int_{-a}^a f(x)\,dx = \mpause[1]{0} \end{talign} \end{itemize} \end{block} \pause\pause \begin{exampleblock}{} \vspace{-1ex} \begin{talign} \int_{-1}^1 f(x)\, dx \mpause[6]{\alert{= 0}} &&\text{where}&& f(x) = \frac{\tan x}{1+x^2 + x^4} \end{talign} \pause The function $f$ is \mpause[4]{\alert{odd} since} \begin{talign} f(-x) \mpause[1]{ = \frac{\sin (-x) / \cos (-x)}{1+(-x)^2 + (-x)^4} } \mpause{ = \frac{-\sin x / \cos x}{1+x^2 + x^4} } \mpause{ = -f(x) } \end{talign} \end{exampleblock} \end{frame}