Vector Calculus · Medium problem

A rectangular loop has vertices $(0,0)$, $(w,0)$, $(w,1)$, $(0,1)$, for some unknown width $w>0$, traversed counterclockwise. The circulation of the field $\mathbf F=(y^2,x^2)$ around this loop, $\displaystyle\oint_C y^2\,dx+x^2\,dy$, equals $6$. Find $w$.