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No answer here but only further forward suggestions with my thoughts on the topic. We can start with any contour C but more conveniently consider a loop with known closed form parametrization. Supposing we start with an "ellipse" contour C written on a unit sphere ( defined by achille hui in SE Math in reply to my question or any Monkey saddle variant) with ...


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tmp = {{0, 0}, {1, 1}, {2, 1}, {3, 2}, {1, 0.5}}; Graphics@ Polygon[tmp] yields this: Its area is Area[Polygon[tmp]] (* 0.583333 *) Have fun!


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tmp = {{0, 0}, {1, 1}, {2, 1}, {3, 2}, {1, 0.5}}; plot = ListLinePlot[tmp, Filling -> Axis]; Area@DiscretizeGraphics[plot] (* 0.833333 *) I figured out how to use Graphics`PolygonUtils`SimplePolygonPartition. It subdivides the polygon from the plot into non self-intersecting, possibly nonconvex polygons, but some of the polygons it creates lie ...


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It turns out that in fact DiscretizeGraphics returns directly a region object, so I can do NIntegrate[1, {x, y} ∈ tt] (* 25.6601 *) or more generally, NIntegrate[x^2 y, {x, y} ∈ tt] Pretty nifty! So one could define a function NImplicitRegion[cond, rg__] := Module[{tt}, tt = ContourPlot[cond, rg, Frame -> False, ContourShading -> ...



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