Answer
\[2\]
Work Step by Step
\[\begin{align}
& \int_{0}^{1}{\int_{0}^{2}{dy}dx} \\
& \text{Find the integral} \\
& \int_{0}^{1}{\int_{0}^{2}{dy}dx}=\int_{0}^{1}{\left[ y \right]_{0}^{2}dx} \\
& =\int_{0}^{1}{\left( 2-0 \right)dx} \\
& =\int_{0}^{1}{2dx} \\
& =\left[ 2x \right]_{0}^{1} \\
& =2 \\
& \text{Using the graph to switch the order of integration} \\
& \int_{0}^{1}{\int_{0}^{2}{dy}dx}=\int_{0}^{2}{\int_{0}^{1}{dx}dy} \\
& =\int_{0}^{2}{\left[ x \right]_{0}^{1}dy} \\
& =\int_{0}^{2}{dy} \\
& =\left[ y \right]_{0}^{2} \\
& =2 \\
\end{align}\]
