Integrate over Regions 

Integrate over basic geometric regions.

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X \[ScriptCapitalR] = Polygon[{{0, 0}, {1, 2}, {4, 0}}];

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X Integrate[Sin[x] Sin[y], {x, y} \[Element] \[ScriptCapitalR]]

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Visualize the region.

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X Plot3D[Sin[x] Sin[y], {x, y} \[Element] \[ScriptCapitalR]]

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Integrate over regions in higher dimensions.

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X Integrate[1, {x[1], x[2], x[3], x[4], x[5]} \[Element] Ball[{0, 0, 0, 0, 0}, r]]

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X NIntegrate[1, {x[1], x[2], x[3], x[4], x[5]} \[Element] Ball[{0, 0, 0, 0, 0}, 1]]

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Integrate over implicit regions.

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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 Integrate[1, {x, y, z} \[Element] \[ScriptCapitalR]]

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X RegionPlot3D[x y z < 1, {x, -5, 5}, {y, -5, 5}, {z, -5, 5}, PlotStyle -> Directive[Yellow, Opacity[0.5]], Mesh -> None]

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Integrate over parametric regions.

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X \[ScriptCapitalR] = ParametricRegion[{Sin[u], Cos[u], u/10}, {{u, 0, 20}}];

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X Integrate[1, {x, y, z} \[Element] \[ScriptCapitalR]]

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X ParametricPlot3D[{Sin[u], Cos[u], u/10}, {u, 0, 20}]

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