Question $8[$Max $10\%]$: Use Green's theorem to find the work done by a force field

vec$(F)(x,y)=($:$-3x^{2}y,3x{y}^2$:$)$

in moving a particle along the piecewise smooth closed curve $C$ from point $(0,0)$ to $(-8,2)$ along $x=-2y^2,$ followed by a line segment from $(-8,2)$ to $(2,2),$ and then along $y=\frac{x^2}{2}$ from the point $(2,2)$ back to the origin $(0,0).$

Remark: The work is given by the line integral of the second kind of force field along the curve.