Example 2 (2004 National High School Competition Question) 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 an interval of the smallest positive period length and the graph of the function $g(x)=\sqrt{a^{2}+1}$ enclose a closed figure whose area is $\qquad$
$\qquad$.
Solve: Fill in $\frac{2 \pi}{a} \sqrt{a^{2}+1}$. Reason: Transform the function $f(x)$ into a familiar form, i.e., $f(x)=\sqrt{a^{2}+1} \sin (a x+\varphi)$, where $\varphi=\arctan \frac{1}{a}$. Its smallest positive period is $\frac{2 \pi}{a}$, and the amplitude is $\sqrt{a^{2}+1}$, as shown in Figure 1-1. The area of the irregular shape formed by the graph of $f(x)$ over one smallest positive period interval $[0, \left.\frac{2 \pi}{a}\right]$ and the graph of $g(x)$, which consists of parts I, II, and III in the shaded area of Figure 1-1, can be found by identifying an equivalent familiar regular shape. By the symmetry of the shape, the irregular shape can be cut and filled (i.e., cut out part of the forest and fill in N) to form a rectangle with length $\frac{2 \pi}{a}$ and width $\sqrt{a^{2}+1}$. Therefore, its area is $\frac{2 \pi}{a} \sqrt{a^{2}+1}$.