Minimizing a line integral

Question

Let $F:\mathbb{R}^2 \to \mathbb{R}^2 $ s.t $$ F(x,y) =\begin{bmatrix} x^2 +y^2 +4y \\ x^3 +xy^2 +2xy\end{bmatrix} $$ Find the simple closed curve that minimizes $\int_C F \cdot d\mathbb{x}$

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I've been stuck on this question and was hoping somebody could help me out. I thought of using the fact that $F$ is of class $C^1$ to use Green's theorem (I'm assuming a curve that minimizes the line integral exists, as the question does not ask us to prove this existence), and I end up with $$\int_C F \cdot d\mathbb{x}=\int_S (3x^2 +y^2 -4)dA$$ where S is a regular region bounded by C. I'm not sure where to go from here though, I see that we are integrating over an ellipse but don't know how to go further. Do I need to differentiate using FTC? Any help/hints is appreciated thank you.

Answer 1

Hint.

A graphical existence confirmation of such a curve:

gr1 = StreamPlot[{x^2 + y^2 + 4 y, x^3 + x y^2 + 2 x y}, {x, -5, 5}, {y, -9, 1}];
sol = NDSolve[{x'[t] == x[t]^2 + y[t]^2 + 4 y[t], y'[t] == x[t]^3 + x[t] y[t]^2 + 2 x[t] y[t], x[0] == 0, y[0] == -0.2}, {x, y}, {t, 0, 10}]
gr2 = ParametricPlot[Evaluate[{x[t], y[t]} /. sol], {t, 0, 10}, PlotRange -> All, PlotStyle -> Red];
Show[gr1, gr2]

Making $F(x,y) = \cases{P(x,y)\\ Q(x,y)}$

solving the ODE $P(x,y) dx + Q(x,y) dy = 0$ gives as solution $V(x,y)= C_0$. It seems to be a conservative system with a limited region in which the orbits are closed curves.