I am very new to this so sorry if it is obvious.
Compute the line integral $\int Fdr $ where $F(x,y)=(x^2y,y^2x)$; $r(t)=(\cos t,\sin t)$; $t\in[0,2\pi]$.
So what I would do is find $r'(t)=(-\sin t,\cos t)$ and $$\int Fdr=\int F(r(t))r'(t)dt=\int (\cos ^2(t)\sin (t), \sin ^2(t)\cos (t))(-\sin (t), \cos (t))dt=\int (-\cos ^2(t)\sin ^2(t) +\sin ^2(t)\cos ^2(t))dt=0$$