b6dc5a…: “Onto the proud parallelogram”

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Let $P$ be a parallelogram (drawn in a counterclockwise fashion when viewed from $z=10$ ) with vertices: $(0,0,0)\quad (3,2,1)\quad (2,3,1) \quad (5,5,2)$ Compute: $\oint _P (x^2-z) \d x + (y^3+x)\d y + y \d z \begin {prompt} = \answer {5} \end {prompt}$