11. (2004 National High School Competition) In the plane rectangular coordinate system $x O y$, the graph of the function $f(x)=a \sin a x+\cos a x(a>0)$ over the interval of its smallest positive period, and the graph of $g(x)=\sqrt{a^{2}+1}$ enclose a closed figure. The area of this figure is
11. $\frac{2 \pi}{a} \sqrt{a^{2}+1} f(x)$ has a maximum value of $\sqrt{a^{2}+1}$, a period of $\frac{2 \pi}{a}$, and by symmetry, the area is $\frac{2 \pi}{a} \sqrt{a^{2}+1}$.