$\begin{aligned} & I=\int_0^a x^2\left(a^2-x^2\right)^{3 / 2} d x \\ & \text { Put } x=a \sin \theta \Rightarrow d x=a \cos \theta d \theta \\ & I=\int_0^{\pi / 2} a^2 \sin ^2 \theta \cdot a^3 \cos ^3 \theta \cdot a \cos \theta d \theta \\ & =a^6 \int_0^{\pi / 2} \sin ^2 \theta \cos ^4 \theta d \theta=a^6 \frac{\Gamma \frac{3}{2} \cdot \Gamma \frac{5}{2}}{2 \cdot \Gamma \frac{8}{2}} \\ & =a^6 \frac{\frac{1}{2} \cdot \sqrt{\pi} \cdot \frac{3}{2} \cdot \frac{1}{2} \cdot \sqrt{\pi}}{2 \cdot 3 \cdot 2 \cdot 1}=\frac{\pi a^6}{32} .\end{aligned}$