11. (2004 National High School Mathematics Competition Question) In the plane rectangular coordinate system $x O y$, the function $f(x)=$ $a \sin a x+\cos a x(a>0)$ over an interval of the smallest positive period length, the area of the closed figure enclosed by its graph and the graph of the function $g(x)=\sqrt{a^{2}+1}$ is $\qquad$ .
11.
$$
\begin{aligned}
f(x) & =a \sin a x+\cos a x=\sqrt{a^{2}+1}\left(\frac{a}{\sqrt{a^{2}+1}} \sin a x+\frac{1}{\sqrt{a^{2}+1}} \cos a x\right) \\
& =\sqrt{a^{2}+1} \sin (a x+\varphi) \cdot\left(\text { where } \varphi=\arctan \frac{1}{a}, a>0\right)
\end{aligned}
$$

From this, we know that the smallest positive period of the function $f(x)$ is $\frac{2 \pi}{a}$, and the amplitude is $\sqrt{a^{2}+1}$. By the symmetry of the closed figure formed by the graphs of $f(x)$ and $g(x)$ (i.e., the shaded part in Figure 1-5), the figure can be divided and complemented into a rectangle with length $\frac{2 \pi}{a}$ and height $\sqrt{a^{2}+1}$. Therefore, its area is $\frac{2 \pi}{a} \sqrt{a^{2}+1}$.