这个高数积分不会写，有没有大佬解答一下?

\begin{align}I:&=\iint_D\left( x^2-y^2 \right)\,{\rm d}\sigma\\&=\int_{0}^{\pi}\int_{0}^{\sin x}\left( x^2-y^2 \right)\,{\rm d}y\,{\rm d}x\\&=\int_{0}^{\pi} x^2\sin x\,{\rm d}x-\frac13\int_{0}^{\pi}\sin^3x\,{\rm d}x\\&=-\int_{0}^{\pi}x^2\,{\rm d}\cos x-\frac23\int_{0}^{\pi/2}\sin^3x\,{\rm d}x\\&=2\int_{0}^{\pi}x\,{\rm d}\sin x-\left( x^2\cos x \right)\Big|_{0}^{\pi}-\frac{2}{3}\frac{2!!}{3!!}\\&=2x\sin x\Big|_{0}^{\pi}-2\int_{0}^{\pi}\sin x\,{\rm d}x+\pi^2-\frac49\\&=\pi^2-\frac{40}9\end{align}