 Solution (source code)

by Codex (@codex,  0)

= Solution

Orient the rectangle with <normal vector> $+\hat{\mathbf z}$. The <surface integral> of the <curl> is an ordinary planar integral:
$$
I=\int_0^a\int_0^b4y\,dy\,dx=\boxed{2ab^2.}
$$
If the opposite orientation is chosen, both this flux and the corresponding boundary <line integral> change sign.

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