smc

Consider the case when $x\ge 0$, $y\ge 0$. $$x^2+y^2=x+y$$ $$(x-\frac{1}{2}{)}^2+(y-\frac{1}{2}{)}^2=\frac{1}{2}$$ Notice the circle intersects the axes at points $(0,1)$ and $(1,0)$. Find the area of this circle in the first quadrant. The area is made of a semi-circle with radius of $\frac{\sqrt{2}}{2}$ and a triangle: $$A=\frac{\pi }{4}+\frac{1}{2}$$ Because of symmetry, the area is the same in all four quadrants. The answer is $\boxed{\pi +2}$