900df3…: “Beyond a summer dodecahedron”

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Consider the vector field $\vec {F} : \R ^2 \to \R ^2$ given by $\vec {F}(x,y) = \vector {3 y - 2 , 3 x + 10 y + 4}$ Let $C$ be the ellipse parameterized by the function $\vec {p} : [0,2\pi ] \to \R ^2$ defined by $\vec {p}(t) = \vector {\cos t, 3\sin t}.$ Compute $\oint _C \vec {F} \dotp \d \vec {p}$ . $\oint _C \vec {F} \dotp \d \vec {p} = \answer {0}.$