= Solution
Starting at $(1,0)$, traverse the right, top, left and bottom sides in that order. For $\mathbf F$, the differential expression is $-y\,dx+x\,dy$. The four <line integral> contributions are respectively $1,1,0,0$, so the total is two.
For $\mathbf G$, the expression is $2xy^2\,dx+2yx^2\,dy$. The right side contributes $\int_0^12y\,dy=1$, and the top side contributes $\int_1^02x\,dx=-1$; the other sides contribute zero. Therefore
$$
\boxed{\oint_{\rm square}\mathbf F\cdot d\mathbf r=2,\qquad
\oint_{\rm square}\mathbf G\cdot d\mathbf r=0.}
$$
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