\begin{frame} \frametitle{Indefinite Integrals} \begin{block}{} The \emph{indefinite integral} \begin{talign} \int f(x)dx \end{talign} is a notation for an antiderivative. That is \begin{talign} \int f(x)dx = F(x) &&\text{means} && F'(x) = f(x) \end{talign} \end{block} \pause \begin{exampleblock}{} For example \begin{talign} \int x^2\,dx = \frac{1}{3}{x^3} + C \end{talign} \end{exampleblock} \medskip\pause \only<-3>{ \begin{alertblock}{} Note the difference between the definite and indefinite integral!\\[.5ex] The definite integral $\int_a^b f(x)\,dx$ is a number.\\[.5ex] The indefinite integral $\int f(x)\,dx$ is a function. \end{alertblock} } \pause The 2nd part of the Fundamental Theorem can be restated as: \begin{block}{} If $f$ is a continuous function then \begin{talign} \int_a^b f(x)\,dx \;=\; \int f(x)\,dx\Big]_a^b \end{talign} \end{block} \vspace{10cm} \end{frame}