\newcommand{\tg}{\mathop{\mathrm{tg}}}{\displaystyle {M_x:=\int_{\cos\theta}^0\int_{x\tg\theta}^{\sqrt{1-x^2}}x\,\hbox{d}y\,\hbox{d}x+\int_0^1\int_0^{\sqrt{1-x^2}}x\,\hbox{d}y\,\hbox{d}x=\int_{\cos\theta}^0 x(\sqrt{1-x^2}-x\tg\theta)\,\hbox{d}x+\int_0^1 x\sqrt{1-x^2}\,\hbox{d}x=\\\int_{\cos\theta}^1x\sqrt{1-x^2}\,\hbox{d}x-{\newcommand{\tg}{\mathop{\mathrm{tg}}}\tg\theta\over 3} x^3\Bigg |_{\cos\theta}^0=-{1\over 2}\int_{\cos\theta}^1(1-x^2)^{1\over 2}\,\hbox{d}(1-x^2)+{\sin\theta\cos^2\theta\over 3}=\\-{1\over 3}(1-x^2)^{3\over 2}\Bigg |_{\cos\theta}^1+{\sin\theta(1-\sin^2\theta)\over 3}={\sin^3\over 3}+{\sin\theta(1-\sin^2\theta)\over 3}={\sin\theta\over 3}}}