If C is the curve $r=(x,y):x^2/a^2+y^2/b^2=1$ then find $\oint_C{r\times dr}$ in an anticlockwise direction
I varied $\theta$ from $0$ to $2\pi$ over the ellipse to get
$r=(acos(\theta),bsin(\theta),0)$
and $dr/d\theta=(-asin(\theta), bcos(\theta),0)$
So that $\oint_C{r\times dr}=\hat{k}\int_0^{2\pi}ab*d\theta$
So the answer is $2\pi(ab)\hat{k}$
Is this correct? The $2\pi$ seems quite arbitrary....