067730…: “Left of a winter cube”

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Consider the vector field $\vec {F} : \R ^2 \to \R ^2$ given by $\vec {F}(x,y) = \vector {-10 x + 2 y - 4 , 2 x - 6 y - 1}$ Let $C$ be the semicircle parameterized by the function $\vec {p} : [0,\pi ] \to \R ^2$ defined by $\vec {p}(t) = \vector {\cos t, \sin t}.$ Compute $\int _C \vec {F} \dotp \d \vec {p}$ . $\int _C \vec {F} \dotp \d \vec {p} = \answer {8}.$