Formula Regions 

Formula regions are defined by some formula, either ImplicitRegion using logical combinations of implicit conditions or ParametricRegion using explicit parametric functions of some parameters. Formula regions provide flexible ways of giving mathematical definitions of a region and can define regions of any geometric dimension and embedded in any dimension.

Regions defined by logical combinations of implicit conditions.

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X \[ScriptCapitalR] = ImplicitRegion[1 <= x^2 + y^2 <= 2, {x, y}];

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X RegionPlot[\[ScriptCapitalR], FrameTicks -> None]

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X Integrate[1, x \[Element] \[ScriptCapitalR]]

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X \[ScriptCapitalR] = ImplicitRegion[x y z <= 1, {{x, -5, 5}, {y, -5, 5}, {z, -5, 5}}];

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X RegionPlot3D[\[ScriptCapitalR], PlotStyle -> Opacity[0.5]]

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X Integrate[1, x \[Element] \[ScriptCapitalR]]

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X \[ScriptCapitalR] = ImplicitRegion[-1 + (-1 + 18 x^2 - 48 x^4 + 32 x^6)^2 + (-1 + 18 y^2 - 48 y^4 + 32 y^6)^2 <= 0, {x, y}];

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X RegionPlot[\[ScriptCapitalR], FrameTicks -> None] // Quiet

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Regions defined by explicit parametric functions.

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X \[ScriptCapitalR] = ParametricRegion[{Cos[\[Theta]], Sin[\[Theta]]}, {{\[Theta], 0, 2 \[Pi]}}];

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X RegionPlot[\[ScriptCapitalR], FrameTicks -> None]

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X \[ScriptCapitalR] = ParametricRegion[{{s^2 t^2, s t^3}, -1 <= s <= 1 && -1 <= t <= 1}, {s, t}];

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X RegionPlot[\[ScriptCapitalR], FrameTicks -> None]

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