I'm asked to compute $$\int_C \vec{F} \cdot d\vec{s}$$ where $\vec{F} = A_0(x\hat{y} - y\hat{x})$, along a circular path, counterclockwise, about the origin with radius 4.
I begin by writing $$ \vec{F} = (-A_0y, A_0x) = (-4A_0sin\theta, 4A_0cos\theta)$$
I run into trouble when I attempt to do the same for $d\vec{s}$. How would I break it up into components such that I can end up integrating with respect to $\theta$?
Let $c: t\mapsto (4\cos t, 4\sin t)$ be a parametrization of the circle. Then $c'(t)=(-4\sin t,4\cos t)$, so $$\begin{align} \int_C\mathbf F\cdot d\mathbf s &= \int_0^{2\pi}(-4A_0\sin t,4A_0\cos t)\cdot(-4\sin t,4\cos t)\;dt \\ &= \int_0^{2\pi}16 A_0 \sin^2t+16 A_0 \cos^2t\;dt = 16A_0\int_0^{2\pi}dt \\ &= 32\pi A_0. \end{align}$$