The question is to find the work done by a force field.
The force field is $F=(-y^2,x)$ and the path is the triangle formed by the points $(0,0),\ (1,0),\ (0,2)$ in an anti-clockwise direction.
We have to find the work done first by using a line integral and then using Green's theorem. I know how to do a normal line integral, but because it's over a shape like this I don't know how to approach it.
Best Answer
You say you know how to do a line integral. The points you give for the triangle are already in anti-clockwise (also called counter-clockwise) order. The "curve" on which you will do the line integral is a triangle, with three sides.
So do three line integrals and add their values. For each pair of points, make a parameterization for the line segment from the first point to the second point. For example, from $(0,0)$ to $(1,0)$ you could use
$$x=t$$ $$y=0$$ $$0\le t\le 1$$
Then do the appropriate line integral over that line segment. From $(0,0)$ to $(1,0)$ you would find $\int_C \mathbf{F}\cdot d\mathbf{s}$ where $C$ is $[0,1]$, $\mathbf{F}=(-y^2,x)=(-(0)^2,t)$, $\mathbf{s}=(t,0)$.
Then do that again from $(1,0)$ to $(0,2)$, then again from $(0,2)$ to $(0,0)$. Add those three integral values and you are done.
Using Green's theorem is another matter, of course.