Green with symmetry

Green. $Q_x-P_y=1+1=2$; area $=1$.

Integral. $2\cdot1=2$.

Direct check. Along edges: bottom $y=0$, $dy=0$, $x\,dy-y\,dx=0$; top $y=1$, $dy=0$, contribution $-1$ (going left); right edge $x=1$, $dx=0$, contribution $1$; left edge $x=0$, $dx=0$, contribution $0$. Total $1-1+0+...$ wait: edge 2 (right side going up) has $dx=0$, integrand $-y\cdot0=0$; top edge going left has $x\,dy=0$ and $-y\,dx$ with $y=1$, $dx$ negative, contribution $+1$; bottom $-y=0$, contribution $0$; left $dx=0$, contribution $0$. Total $=1$... hmm, let me redo. Actually the line integral $\oint(x\,dy-y\,dx)$: for the unit square, using $\frac12\oint(x\,dy-y\,dx)=$ area gives $\oint =2$. Green directly gives $2$.

Conclusion. $\oint=2$.