Answer
$\frac{9}{2}\pi$
Work Step by Step
\[\begin{align}
& \int_{-3}^{3}{\sqrt{9-{{x}^{2}}}dx} \\
& \text{The integral represents the area of a semicircle of radius 3, see figure} \\
& \text{below}\text{.} \\
& \text{The area of the circle is given by} \\
& A=\pi {{r}^{2}} \\
& \text{Let }r=4 \\
& A=\pi {{\left( 3 \right)}^{2}} \\
& A=9\pi \\
& \text{Then } \\
& \text{The area of the integral is } \\
& \frac{1}{2}A=\frac{1}{2}\left( 9\pi \right) \\
& \frac{1}{2}A=\frac{9}{2}\pi \\
\end{align}\]
