$\frac{\mathrm{x}}{2\mathrm{a}-\mathrm{x}}\ge 0\Rightarrow \mathrm{x}\in [0,2\mathrm{a});{\mathrm{k\pi a}}^2=2{\int }_0^{2\mathrm{a}} x\sqrt{\frac{\mathrm{x}}{2\mathrm{a}-\mathrm{x}}}\mathrm{dx}$

$x=2a {\sin }^2\theta \Rightarrow dx=4a\sin \theta \cos \theta d\theta$

$$\begin{gathered} ;{\mathrm{k\pi a}}^2=2{\int }_0^{\frac{\mathrm{\pi }}{2}} 2a{\sin }^2\theta \sqrt{\frac{2{\mathrm{asin}}^2\mathrm{\theta }}{2\mathrm{a}-2{\mathrm{asin}}^2\mathrm{\theta }}}\left(4a\sin \theta \cos \theta d\theta \right) \\ \Rightarrow {\mathrm{k\pi a}}^2=16a^2{\int }_0^{\frac{\mathrm{\pi }}{2}}{\sin }^4\theta d\theta \end{gathered}$$

$;{\mathrm{k\pi a}}^2=16{\mathrm{a}}^2\left(\frac{3}{4}\cdot \frac{1}{2}\cdot \frac{\mathrm{\pi }}{2}\right)\Rightarrow \mathrm{k}=3;\left\{\mathrm{k}\right\}=0$