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Mathematical reference only; not implementation validation or an endorsement of CalculatorX.", "kind": "reference", "publisher": "Wellesley-Cambridge Press; hosted by MIT OpenCourseWare" }, { "title": "Gilbert Strang, Calculus — Chapter 15: Vector Calculus (MIT OpenCourseWare)", "section": "§15.3 Green’s Theorem", "url": "https://ocw.mit.edu/courses/res-18-001-calculus-fall-2023/mitres_18_001_f17_ch15.pdf", "supports": "The planar area formula A = (1/2) ∮(x dy − y dx). For straight polygon edges it gives the shoelace expression; rectangular/square areas and trapezoid area (B+b)h/2 follow by applying it to the vertices. For x=a cos t, y=b sin t it gives ellipse area πab. Perimeters of these polygons are sums of side lengths. These formulas are derived applications of the theorem. 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