Greens Theorem to Evaluate curve $2\lvert x\rvert + \lvert y\rvert = 1$

Question

Trying to evaluate the function $(x^3 + y^3)\,\mathrm{d}x + (x^3 - y^3)\,\mathrm{d}y$ over the curve $2\lvert x\rvert + \lvert y\rvert = 1$ using greens theorem. The correct answer is $-3/8$ which I get only if I double integrate over one out of the four of the triangles. I thought the answer multiplies by four because of symmetry?

Answer 1

The curl $Q_x-P_y=3(x^2-y^2)$ is symmetric with respect to $x$ and $y$. Therefore it is sufficient to compute $$4\int_0^{1/2}\int_0^{1-2x}3(x^2-y^2)\>dy\>dx\ ,$$ and the result is indeed $-{3\over8}$.