Answer
The integral is $0$ irrespective of the curve $C$.
Work Step by Step
The tangential form for Green's Theorem can be computed as:
The counterclockwise Circulation is : $\oint_C F \cdot T ds= \iint_{R} (\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}) dx dy $
$\implies \oint_C x^3 \ dx -y^3 \ dy= \iint_{R} (\dfrac{\partial (-y^3) }{\partial x}-\dfrac{\partial (x^3) }{\partial y}) \ dx \ dy $
$\implies \oint_C x^3 \ dx -y^3 \ dy= \iint_{R} [0-0] \ dx \ dy$
$\implies \oint_C x^3 \ dx -y^3 \ dy=\oint_C x^3 \ dx -y^3 \ dy=0$
This means that the integral is $0$ irrespective of the curve $C$.
