Solution
= Solution
Parametrize the directed segment by $\mathbf r(t)=(t,t,0)$, $0\leq t\leq1$. Its tangent is $(1,1,0)$, while the restricted <vector field> is $(t^3,t^3,0)$. The <line integral> is therefore
$$
\boxed{\int_0^1\mathbf G(\mathbf r(t))\cdot\mathbf r'(t)\,dt
=\int_0^12t^3\,dt=\frac12}.
$$
Alternatively, $\mathbf G=\nabla(x^2y^2/2)$, so the <fundamental theorem for line integrals> gives the same endpoint difference.