\begin{frame}
  \frametitle{Comparison Properties of the Definite Integral}

  \begin{block}{}
    \begin{itemize}
    \pause
      \item If $f(x) \ge 0$ for all $a \le x \le b$, then
        \begin{malign}
          \int_a^b f(x)dx \ge 0
        \end{malign}
    \pause\smallskip
      \item If $f(x) \ge g(x)$ for all $a \le x \le b$, then
        \begin{malign}
          \int_a^b f(x)dx \;\ge\; \int_a^b g(x)dx
        \end{malign}
    \pause\smallskip
      \item If $m \le f(x) \le M$ for all $a \le x \le b$, then
        \begin{malign}
          m(b-a) \;\le\; \int_a^b f(x)dx \;\le\; M(b-a)
        \end{malign}
    \end{itemize}
  \end{block}
  \pause
  \begin{exampleblock}{}
    Use the last property to estimate $\int_0^1 e^{-x^2}dx$.
    \pause\smallskip
    
    The function $e^{-x^2}$ is decreasing on $[0,1]$.
    \pause\smallskip
    
    Thus on $[0,1]$: maximum is \pause $f(0) = 1$\pause,
    and minimum \pause $f(1) = e^{-1}$.\pause
    \begin{malign}
      e^{-1}(1-0) = e^{-1} \quad\le\quad \int_0^1 e^{-x^2}dx \quad\le\quad 1 = 1(1-0)  
    \end{malign}\vspace{-.2ex}
  \end{exampleblock}
\end{frame}