 Solution

Set $P=10x^4-2x{y}^3$ and $Q=-3x^2{y}^2$. Their partial derivatives satisfy $P_y=-6x{y}^2=Q_x$ throughout ${\mathbb{R}}^2$. The vector field is smooth on the entire simply connected plane, so zero curl proves path independence.

Choose the broken straight path $(0,0)\to (2,0)\to (2,1)$. Along its horizontal segment $y=0$ and $dy=0$, givingAlong its vertical segment $x=2$ and $dx=0$, givingTherefore the line integral is