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geometry - Volume of solids of revolution through two equations



I would like to get the area formed by two equations and the volume of the solid formed by this area around the Y axis.


Solve[x^2==6x-2x^2,x]



{{x->0},{x->2}}



Plot[{x^2,6x-2x^2},{x,0,2}]

enter image description here


I tried to use this idea but did not succeed;


Intersection Volume



Answer



I propose another way. You can use DiscretizeRegion to create a region from the ParametricRegion. Then, calculate the volume:


\[ScriptCapitalR]1 = 

 DiscretizeRegion[
 ParametricRegion[{{x, y, z},6 Sqrt[x^2 + y^2] - 2 (x^2 + y^2) > z && x^2 + y^2 <= 4 &&x^2 + y^2 <= z && z >= 0}, {{x, -4, 4}, {y, -4, 4}, {z, 0, 5}}],
 MaxCellMeasure -> #, ImageSize -> Small] & /@ {Automatic,0.1, 0.01,0.0005};
 vols1 = Volume[\[ScriptCapitalR]1[[#]]] & /@ Range[4];
 Row@(Labeled[\[ScriptCapitalR]1[[#]],Style["Volume = " <> ToString[vols1[[#]]], 12, FontFamily -> "Times New Roman"], Top]&/@ Range[4])

enter image description here


or use BoundaryDiscretizeRegion:


\[ScriptCapitalR]2 = 
 BoundaryDiscretizeRegion[

 ParametricRegion[{{x, y, z},6 Sqrt[x^2 + y^2] - 2 (x^2 + y^2) > z && x^2 + y^2 <= 4 &&x^2 + y^2 <= z && z >= 0}, {{x, -4, 4}, {y, -4, 4}, {z, 0, 5}}],
 MaxCellMeasure -> #, ImageSize -> Small] & /@ {Automatic,0.1, 0.01,0.0005};
 vols2 = Volume[\[ScriptCapitalR]2[[#]]] & /@ Range[4];
 Row@(Labeled[\[ScriptCapitalR]2[[#]],Style["Volume = " <> ToString[vols2[[#]]], 12, FontFamily -> "Times New Roman"], Top]&/@ Range[4])

enter image description here


In both cases, the value of the calculated volume, and obviously the appearance, depends on the value of the option MaxCellMeasure, as can be deduced from the plots.


It seems that BoundaryDiscretizeRegion has a greater accuracy.


The volume can be calculated directly from ParametricRegion, which is more exact, and compared to previous values:


NestList[N,Volume[ParametricRegion[{{x, y, z}, 

6 Sqrt[x^2 + y^2] - 2 (x^2 + y^2) > z && x^2 + y^2 <= 4 && x^2 + y^2 <= z && z >= 0},
{{x, -4, 4}, {y, -4, 4}, {z, 0, 5}}]],1]

(*{8 \[Pi], 25.1327}*)

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