Let $C$ be a circunference of radius $a$ , in the plane $2x+2y+z=4$, centered at the point (1,2,-2). If $F(x,y,z)=(y-x,z-x,x-y)$, determine the value for $a$ such that $\oint_C F \cdot dr = -8\pi/3$.
I did the easy part, used the Stokes' Theorem to find that the line integral was equal $-8\iint \,dx\,dy$. However, I could not find the limits of integration.
$$\int_c F\mathrm{d}r=\int_c (y-x)\mathrm{d}x+(z-x)\mathrm{d}y+(x-y)\mathrm{d}z=\int-2\mathrm{d}x\mathrm{d}y-\mathrm{d}z\mathrm{d}x-2\mathrm{d}y\mathrm{d}z=\int-2\cos\gamma-\cos\beta-2\cos\alpha\mathrm{d}S$$ $$\vec{n}=(2,2,1) ,\cos\alpha=2/3 ,\cos\beta=2/3 , \cos\gamma=1/3$$ $$\int_c F\mathrm{d}r=\int -\frac{8}{3}\mathrm{d}S=-\frac{8}{3}\pi a^2$$ $$a=1$$