$\begin{array}{l}
\sin x + \sqrt 3 \cos x = \sqrt 2 \,\,\, \Rightarrow \frac{1}{2}\sin x + \frac{{\sqrt 3 }}{2}\cos x = \frac{{\sqrt 2 }}{2}\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \cos \frac{\pi }{3}\sin x + \sin \frac{\pi }{3}\cos x = \frac{{\sqrt 2 }}{2}\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \sin (\frac{\pi }{3} + x) = \sin \frac{\pi }{4}
\end{array}$

$\left\{ \begin{array}{l}
\frac{\pi }{3} + x = 2k\pi  + \frac{\pi }{4}\\
\frac{\pi }{3} + x = 2k\pi  + \pi  - \frac{\pi }{4}
\end{array} \right.\,\,\,\,\,\,\, \Rightarrow \left\{ \begin{array}{l}
x = 2k\pi  + \frac{\pi }{4} - \frac{\pi }{3}\\
x = 2k\pi  + \pi  - \frac{\pi }{4} - \frac{\pi }{3}
\end{array} \right.\,\,\,\,\,\, \Rightarrow \left\{ \begin{array}{l}
x = 2k\pi  - \frac{\pi }{{12}}\\
x = 2k\pi  + \frac{{5\pi }}{{12}}
\end{array} \right.\,\,$

$x = 2k\pi  - \frac{\pi }{{12}}\,\,\,\,\,\, \Rightarrow \left\{ \begin{array}{l}
k =  \circ \,\,\,\,\, \to x =  - \frac{\pi }{{12}}\\
k = 1\,\,\,\,\,\, \to x = 2\pi  - \,\frac{\pi }{{12}}
\end{array} \right.$

$x = 2k\pi  + \frac{{5\pi }}{{12}}\,\,\,\,\,\, \Rightarrow k =  \circ \,\,\,\,\, \to x = \frac{{5\pi }}{{12}}$

$S =  - \frac{\pi }{{12}} + 2\pi  - \,\frac{\pi }{{12}} + \frac{{5\pi }}{{12}} = 2\pi  + \frac{{3\pi }}{{12}} = 2\pi  + \frac{\pi }{4} = \frac{{9\pi }}{4}$