${\int }_0^{\mathrm{\π}\/4}\left(\cos \left(x\right)-\sin \left(x\right)\right) \mathit{\ⅆ}x$ = $\sqrt{2}-1$ =  

$\stackrel{\&conjugate0;}{x}\=\frac{1}{A}{\int }_0^{\mathrm{\π}\/4}x \left(\cos \left(x\right)-\sin \left(x\right)\right) \mathit{\ⅆ}x$ = $\frac{\mathrm{\π} \sqrt{2}-4}{4\left(\sqrt{2}-1\right)}$ 

  and$$

$\stackrel{\&conjugate0;}{y}$ = $\frac{1}{A}{\int }_0^{\mathrm{\π}\/4}\left({\cos }^2\left(x\right)-{\sin }^2\left(x\right)\right)\/2 \mathit{\ⅆ}x$ = $\frac{1\+\sqrt{2}}{4}$

Calculate $A$, the area of region $R_x$ 

${\int }_0^{\mathrm{\π}\/4}\left(\cos \left(x\right)-\sin \left(x\right)\right) \mathit{\ⅆ}x$ = $\sqrt{2}-1$$\stackrel{\text{assign to a name}}{\to }$$A$

Calculate $\stackrel{\&conjugate0;}{x}$ 

$\frac{1}{A}{\int }_0^{\mathrm{\π}\/4}x \left(\cos \left(x\right)-\sin \left(x\right)\right) \mathit{\ⅆ}x$ = $\frac{-1\+\frac{1}{4}\sqrt{2}\mathrm{\π}}{\sqrt{2}-1}$$\stackrel{\text{at 10 digits}}{\to }$$0.2673034979$

Calculate $\stackrel{\&conjugate0;}{y}$ 

$\frac{1}{A}{\int }_0^{\mathrm{\π}\/4}\left({\cos }^2\left(x\right)-{\sin }^2\left(x\right)\right)\/2 \mathit{\ⅆ}x$ = $\frac{1}{4\left(\sqrt{2}-1\right)}$$\stackrel{\text{at 10 digits}}{\to }$$0.6035533912$

$\frac{1}{4\left(\sqrt{2}-1\right)}$$\stackrel{\text{rationalize}}{\=}$$\frac{1}{4}\+\frac{1}{4}\sqrt{2}$