Iy = \(\iint\limits_{R}\mathrm x^2dA\)
\(=\int\limits^\pi_0\left(\int^{\sin \mathrm x}_0\mathrm x^2dy\right)d\mathrm x\)
\(=\int\limits^\pi_0\mathrm x^2(y)^{\sin \mathrm x}_0 d\mathrm x\)
\(=\int\limits^{\pi}_0 \mathrm x^2\sin \mathrm x \,d\mathrm x\)
\(=\left[\mathrm x^2\int \sin \mathrm x\,d\mathrm x - \int\left(2\mathrm x\int \sin \mathrm x d\mathrm x\right)d\mathrm x\right]^\pi_0\)
\(=\left[-\mathrm x^2\cos\mathrm x+\int 2\mathrm x\cos\mathrm x\,d\mathrm x\right]^\pi_0\)
\(=[-\mathrm x^2\cos\mathrm x+2\mathrm x\)\(\int \cos\mathrm x\,d\mathrm x-\int 2(\int \cos\mathrm x \,d\mathrm x)d\mathrm x]^\pi_0\)
\(=[-\mathrm x^2\cos\mathrm x+2\mathrm x\sin \mathrm x-\int 2\sin \mathrm x d\mathrm x]^\pi_0\)
\(=[-\mathrm x^2\cos\mathrm x+2\mathrm x \sin \mathrm x+2\cos\mathrm x]^\pi_0\)
\(=(-\pi^2\cos\pi +2\pi \sin \pi+2\cos\pi)\) \(-(2\cos 0)\)
\(=\pi^2-2-2\) \((\because \cos \pi =\lambda, \cos 0 =1\) & \(\sin \pi =0)\)
\(=\pi^2-4\)
