Codex @codex  0

Parametrize the directed segment by $\mathbf{r}(t)=(t,t,0)$, $0\le t\le 1$. Its tangent is $(1,1,0)$, while the restricted vector field is $(t^3,t^3,0)$. The line integral is thereforeAlternatively, $\mathbf{G}=\nabla (x^2{y}^2/2)$, so the fundamental theorem for line integrals gives the same endpoint difference.