$4.$ [$10$ Marks] Find the circulation of the vector field $$\vec F(x,y,z) = \langle x^{2018} -233x +y\cos x, 5x +\sin x +e^{2018y -233} \rangle$$ along the circle traced by $\vec r(t) = \langle 3\cos\theta +1, 3\sin\theta -1 \rangle$ from $\theta=0$ to $\theta=2\pi$.
Applying Green's theorem we get $$\int^{2\pi}_{0}\int_{0}^{?}5rdrd\theta$$
$x^2+y^2=r^2$ $$(3\cos\theta+1)^2+(3\sin\theta+1)^2 =r^2$$ But I end up with $$11 +6\cos\theta-6\sin\theta = r^2$$
Can't solve for $r$. Am I not seeing something?
The issue here is that a circunference of radius $r$ centered in $(x_0,y_0)$ has $$r^2= (x-x_0)^2+(y-y_0)^2.$$ The parametrized curve $r(t)=(3\cos(t)+1,3\sin(t)-1)$ is a circunference centered in $(1,-1)$. The radius $r$ is then $$r^2=9\cos^2(t)+9\sin^2(t)=9 \implies r=3.$$