The value of the expression $\frac{{\displaystyle \underset{0}{\overset{a}{\int }}{x}^4\sqrt{a^2-x^2}dx}}{{\displaystyle \underset{0}{\overset{a}{\int }}{x}^2\sqrt{a^2-x^2}dx}}$

$$\begin{gathered} \frac{{\displaystyle \underset{0}{\overset{\mathrm{a}}{\int }}{\mathrm{x}}^4\sqrt{{\mathrm{a}}^2-{\mathrm{x}}^2 }\mathrm{dx}}}{{\displaystyle \underset{0}{\overset{\mathrm{a}}{\int }}{\mathrm{x}}^2\sqrt{{\mathrm{a}}^2-{\mathrm{x}}^2 }\mathrm{dx}}} \\ \begin{array}{rcl} & & \end{array}\begin{array}{rcl} & & \end{array}\begin{array}{rcl} & & \end{array}\begin{array}{rcl} & & \end{array}\begin{array}{rcl} & & \end{array}\begin{array}{rcl} & & \end{array}\begin{array}{rcl} & & \mathrm{Put}\end{array}\begin{array}{rcl} & & \end{array}\begin{array}{rcl} & & \end{array}\begin{array}{rcl} & & \mathrm{x}\end{array}\begin{array}{rcl} & =& \end{array}\begin{array}{rcl} & & \mathrm{a}\end{array}\begin{array}{rcl} & & \end{array}\begin{array}{rcl} & & \mathrm{sin\theta }\end{array}\begin{array}{rcl} & & \end{array}\begin{array}{rcl} & & \end{array}\begin{array}{rcl} & & \end{array}\begin{array}{rcl} & & \mathrm{dx}\end{array}\begin{array}{rcl} & =& \end{array}\begin{array}{rcl} & & \mathrm{a}\end{array}\begin{array}{rcl} & & \end{array}\begin{array}{rcl} & & \mathrm{cos\theta }\end{array}\begin{array}{rcl} & & \end{array}\begin{array}{rcl} & & \mathrm{d\theta }\end{array}\begin{array}{rcl} & & \end{array} \end{gathered}$$
$\begin{array}{rcl}\mathrm{I}& =& \frac{{\int }_0^{{\displaystyle \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}} {\mathrm{a}}^4 {\sin }^4\mathrm{\theta } \mathrm{a} \mathrm{cos\theta }.\mathrm{a} \mathrm{cos\theta } \mathrm{d\theta }}{{\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} {\mathrm{a}}^2 {\sin }^2\mathrm{\theta } \mathrm{a} \mathrm{cos\theta } \mathrm{a} \mathrm{cos\theta } \mathrm{d\theta }}\end{array}$
$\begin{array}{rcl} & =& \mathrm{a}{ }^2\frac{{\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} {\sin }^4\mathrm{\theta } {\cos }^2\mathrm{\theta } \mathrm{d\theta }}{{\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} {\sin }^2\mathrm{\theta } {\cos }^2\mathrm{\theta } \mathrm{d\theta }} \\ & =& \mathrm{a}{ }^2 \frac{\left[{\displaystyle \frac{3.1.1}{6.4.2}·\frac{\mathrm{\pi }}{2}}\right]}{{\displaystyle \frac{1.1}{4.2}·\frac{\mathrm{\pi }}{2}}} =\frac{{\mathrm{a}}^2}{2}\end{array}$