China Mathematical Competition (Hainan)

We rewrite function $f(x)$ as $f(x)=\sqrt{a^2+1}\sin (ax+\phi )$, where $\phi =\arctan \frac{1}{a}$. Its least positive period is $\frac{2\pi }{a}$, and its amplitude is $\sqrt{a^2+1}$. By symmetry of the figure enclosed by the graphs of the functions $f(x)$ and $g(x)$, we can change the figure into a rectangle with length $\frac{2\pi }{a}$ and width $\sqrt{a^2+1}$ using the cut-and-paste method.

Therefore its area is $\frac{2\pi \sqrt{a^2+1}}{a}$.