\begin{frame} \frametitle{Properties of the Definite Integral} \begin{exampleblock}{} Assume $\int_0^{10} f(x) dx = 17$ and $\int_0^{8} f(x) dx = 12$, find $\int_8^{10} f(x) dx$. \pause \begin{talign} \int_0^{8} \!\!f(x) dx + \int_8^{10} \!\!\!f(x) dx = \int_0^{10} \!\!\!f(x) dx \implies \mpause[1]{\int_8^{10} \!\!\!f(x) dx = 17-12=5} \end{talign} \end{exampleblock} \pause\pause \begin{exampleblock}{} Use the properties of integrals to evaluate: \begin{talign} \int_0^1 (4+3x^2)dx &= \mpause[1]{ \int_0^1 4\;dx + \int_0^1 3x^2 dx} \\ \mpause{&= 4 + 3\int_0^1 x^2 dx} \mpause{}\\ \mpause{& = 4 + 3\frac{1}{3} = 5} \end{talign} \pause\pause\pause We have already seen that \begin{talign} \int_0^1 x^2dx = \frac{1}{3} \end{talign} \end{exampleblock} \end{frame}