Answer
\[\frac{9}{2}\]
Work Step by Step
\[\begin{gathered}
\int_1^4 {\left( {1 - x} \right)\left( {x - 4} \right)} dx \hfill \\
\hfill \\
{\text{simplify the}}\,\,{\text{integrand}} \hfill \\
\hfill \\
\int_1^4 {\left( {5x - {x^2} - 4} \right)} dx \hfill \\
\hfill \\
{\text{integrating}} \hfill \\
\hfill \\
\left( {\frac{{5{x^2}}}{2} - \frac{{{x^3}}}{3} - 4x} \right)_1^4 \hfill \\
\hfill \\
{\text{Use the fundamental theorem of calculus}} \hfill \\
\hfill \\
\left( {\frac{{5{{\left( 4 \right)}^2}}}{2} - \frac{{{{\left( 4 \right)}^3}}}{3} - 4\left( 4 \right)} \right) - \left( {\frac{{5{{\left( 1 \right)}^2}}}{2} - \frac{{{{\left( 1 \right)}^3}}}{3} - 4\left( 1 \right)} \right) \hfill \\
\hfill \\
{\text{Simplify}} \hfill \\
\hfill \\
\frac{8}{3} + \frac{{11}}{6} \hfill \\
\hfill \\
\frac{9}{2} \hfill \\
\end{gathered} \]