Compute $\int \limits_{C} F.dr$ for $F(x,y)=(\sin(x^3)-xy, y^3\sin(y)+x)$ and $C$ is the curve given by $$2x^2+3y^2=2y$$ while going clockwise.
Having typed this curve on WFA, it turns out to be an ellipse but I have no knowledge of conics anymore... Can someone help me start to parameterise this and find the limits of integration please.
$$2x^2+3y^2=2y$$ $$2x^2+3\left(y-\frac{1}{3}\right)^2=\frac{1}{3}$$
$x=\frac{\rho\cos(\theta)}{\sqrt2}$
$y=\frac{1}{3}+\frac{\rho\sin(\theta)}{\sqrt3}$ $$2x^2+3(y-\frac{1}{3})^2=\rho^2=\frac{1}{3}$$ $\rho=\frac{1}{\sqrt3}$
The parametric equation of the ellipse is : $$x=\frac{\cos(\theta)}{\sqrt6}$$
$$y=\frac{1}{3}+\frac{\sin(\theta)}{3}$$ Then, the integral can be transform to an integral which variable is $\theta$ (from $0$ to $2\pi$)