Answer
$4\pi$
Work Step by Step
\[\begin{align}
& \int_{-4}^{0}{\sqrt{16-{{x}^{2}}}dx} \\
& \text{The integral represent half semicircle of radius 4, see figure} \\
& \text{below}\text{.} \\
& \text{The area of the circle is given by} \\
& A=\pi {{r}^{2}} \\
& \text{Let }r=4 \\
& A=\pi {{\left( 4 \right)}^{2}} \\
& A=16\pi \\
& \text{Then } \\
& \text{The area of the integral is } \\
& \frac{1}{4}A=\frac{1}{4}\left( 16\pi \right) \\
& \frac{1}{4}A=4\pi \\
\end{align}\]
