93fe0d…: “Below the winter river”

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Consider the vector field $\vec {F} : \R ^2 \to \R ^2$ given by $\vec {F}(x,y) = \vector {y, -x}.$ Let $C$ be the circle parameterized by the function $\vec {p} : [0,2\pi ] \to \R ^2$ defined by $\vec {p}(t) = \vector {2\cos t, 2\sin t}.$ Compute $\int _C \vec {F} \dotp \d \vec {p}$ . $\int _C \vec {F} \dotp \d \vec {p} = \answer {-8\pi }.$