\int\limits_{\{(x,y):x^2+y^2\leq x+y\}}(-x^2-y^2+x+y)\,dx\,dy=\\
=\int\limits_{\{(x,y):(x-1/2)^2+(y-1/2)^2\leq 1/2\}}(-(x-1/2)^2-(y-1/2)^2+1/2)\,dx\,dy=\\
=\int_0^{1/\sqrt{2}}\int_0^{2\pi}(-r^2+1/2)r\,dr\,d\varphi=\\
=2\pi\int_0^{1/\sqrt{2}}(-r^2+1/2)r\,dr