ff3a63…: “Throughout the rainy eyes”

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Let $T$ be a triangle (drawn in a counterclockwise fashion when viewed from $z=10$ ) with vertices: $(2,4,1)\quad (2,2,1) \quad (5,3,2)$ Compute: $\oint _T (z-x) \d x + (x+y+z)\d y + (z^2-x) \d z \begin {prompt} = \answer {4} \end {prompt}$