Answer
\[2\]
Work Step by Step
\[\begin{align}
& \int_{1}^{2}{\int_{2}^{4}{dx}dy} \\
& \text{Find the integral} \\
& \int_{1}^{2}{\int_{2}^{4}{dx}dy}=\int_{1}^{2}{\left[ x \right]_{2}^{4}dy} \\
& =\int_{1}^{2}{\left( 4-2 \right)dy} \\
& =\int_{1}^{2}{2dy} \\
& =\left[ 2y \right]_{1}^{2} \\
& =2\left( 2 \right)-2\left( 1 \right) \\
& =2 \\
& \text{Using the graph to switch the order of integration} \\
& \int_{1}^{2}{\int_{2}^{4}{dx}dy}=\int_{2}^{4}{\int_{1}^{2}{dy}dx} \\
& =\int_{2}^{4}{\left[ y \right]_{1}^{2}dx} \\
& =\int_{2}^{4}{\left( 2-1 \right)dx} \\
& =\int_{2}^{4}{dx} \\
& =4-2 \\
& =2 \\
\end{align}\]
