Verify Green's theorem on the plane for the vector field $\ \mathbf F =3xy\mathbf i -x\mathbf j$ along the circle$\ c$ of radius$\ a$ centred at $\ (x,y)=(a,0)$ with counterclockwise direction.
Clearly the equation of the circle is
$\ y^2+(x-a)^2=a^2$,
rearranging and using polar coordinates yields
$\ 0\le r\le 2a\cos\theta$, and$\ -\frac \pi2\le \theta \le \frac \pi2$, correct?
Using Green's theorem I then arrive at
$\ \iint(-1-3x)dxdy$,
and using polar coordinates
$\ \int_{-\frac \pi2}^{\frac \pi2}\int_0^{2a\cos\theta}(-r-3ar\cos\theta) drd\theta$
Solving this integral I get$\ -4\pi a^2$.
I am struggling to solve the integral as a work integral (which I'll need to do in order to prove Green's theorem).
Could somebody please verify my answer using Green's theorem, and explain how I can set up a work integral in order to prove my answer via Green's theorem?
Thanks!
Edit: Please avoid using a parameterisation method for the solution; I have not been taught this and don't completely understand, sorry!