Green's theorem Area formula additional cases

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In this thread, the OP states 3 area formulas and asks for proof. The answer given uses 3 specific cases where ∂Q/∂x − ∂P/∂y=1. I see why these 3 cases give the area formulas, but what about other cases where the difference of partial derivatives = 1, but the specific values of P and Q differ from the 3 cases given? Could those cases produce area computations different from the ones given?