Answer
\[9\]
Work Step by Step
\[\begin{align}
& \int_{0}^{9}{\int_{\sqrt{x}}^{3}{dy}dx} \\
& \text{Find the integral} \\
& \int_{0}^{9}{\int_{\sqrt{x}}^{3}{dy}dx}=\int_{0}^{9}{\left[ y \right]_{\sqrt{x}}^{3}dx} \\
& =\int_{0}^{9}{\left( 3-\sqrt{x} \right)dx} \\
& =\left[ 3x-\frac{2}{3}{{x}^{3/2}} \right]_{0}^{9} \\
& =\left[ 3\left( 9 \right)-\frac{2}{3}{{\left( 9 \right)}^{3/2}} \right]-\left[ 3\left( 0 \right)-\frac{2}{3}{{\left( 0 \right)}^{3/2}} \right] \\
& =9 \\
& \text{Using the graph to switch the order of integration} \\
& \int_{0}^{9}{\int_{\sqrt{x}}^{3}{dy}dx}=\int_{0}^{3}{\int_{0}^{{{y}^{2}}}{dx}dy} \\
& =\int_{0}^{3}{{{y}^{2}}dy} \\
& =\left[ \frac{1}{3}{{y}^{3}} \right]_{0}^{3} \\
& =\frac{1}{3}{{\left( 3 \right)}^{3}} \\
& =9 \\
\end{align}\]
