3fc161…: “Across the interesting quadrilateral”

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Let $\vec {F}:\R ^2\to \R ^2$ be roughly described by the following table of values:

Let $\vec {p}(t)$ be the vector-valued function describing the line $\l$ running from the point $(1,-1)$ to $(-1,1)$ as $t$ runs from $0$ to $1$ . Setting $\d t = 1/3$ , estimate: $\int _\l \vec {F}\dotp \d \vec {p}$ $\vec {p}(t) = \vector {\answer {1-2t},\answer {-1+2t}}$ $\int _\l \vec {F}\dotp \d \vec {p} \approx \answer {-2/3}$