No AM-GM here!

We have:

$\begin{aligned} \displaystyle \int_0^{2\pi} \sqrt{a^2\sin^2x+9\cos^2x} dx&\ge\int_0^{2\pi} \sqrt{9\cos^2x}dx\\ &=\int_0^{2\pi} 3|\cos x|dx\\ &=3\int_0^{\frac{\pi}{2}}\cos x dx-3\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}}\cos x dx+3\int_{\frac{3\pi}{2}}^{2\pi}\cos x dx\\ &=12 \end{aligned}$

This can be transformed into an ellipse equation. The integral refers to total length of ellipse The minimal length happens when a=0. Length of ellipse becomes 2*(2 * 3)=12

Differentiate the above equation w.r.t. to $'a'$ . And then using integral under the derivative we see it is minimum at $a=0$ .