Why function in multivariable integration limits: "not machine-sized #" ?

Questions
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1

RegionPlot3D[Abs[Sin[z]] r, {r, 0, Sqrt[Cos[2 [Theta]]]}, {[Theta], 0, [Pi]/4}, {z, 0, 2 [Pi]}, PlotRange → Automatic]

Trying to plot this returns error: ““Limiting value Sqrt[Cos[2\[Theta]]] in {r,0,Sqrt[Cos[2\[Theta]]]} \is not a machine-sized real number””

Looking through all the different 3DPlot types…did not find any examples of functions in the limits of multivariable integration…?

I’m converting from Maple10, which does not have any problem plotting the above. Mathematica11 will integrate a multivariable with the function in the limits…but no-go on plotting?

2

In my opinion, there are two errors in your code. First, the interval for r must be written correctly in the form specified by the documentation:

RegionPlot3D.png
RegionPlot3D.png574×524 43.5 KB

In your code r_{max} must be a number, but not a variable expression dependent of \Theta and z.

Second, the expression for RegionPlot3D[] must be an inequality. So, by providing these two corrections and fixing r_{max} to value 1.2 Sqrt[\pi/4], the code

RegionPlot3D[
 Abs[Sin[z]] <= r, {r, 0, 1.2 Sqrt[\[Pi]/4]}, {\[Theta], 0, \[Pi]/
  4}, {z, 0, 2 \[Pi]}, PlotRange -> Automatic]

generates the following graph:

Graph3D.png
Graph3D.png698×472 44.4 KB

Sure, the code becomes more interesting if we use the function Manipulate[]:

Manipulate[
 RegionPlot3D[
  Abs[Sin[z]] <= r, {r, 0, rmax}, {\[Theta], 0, \[Pi]/4}, {z, 0, 
   2 \[Pi]}, PlotRange -> Automatic], {rmax, 0.1, 3}]
3

Thanks Valeriu,
Actually I was able to get the result I was looking for with:

RevolutionPlot3D[{(Sqrt[Cos[2*\[Theta]]])*Sin[t], t}, {t, 0, 2 Pi}, {\[Theta], -Pi/4, Pi/4}]`

RevPlot3D.jpg
RevPlot3D.jpg340×403 14.8 KB

This is the exact same code as Maple10 plot3d cylindrical coordinates, so perhaps RevolutionPlot3D could be called cylindrical coordinates.

4

It was very difficult from your initial code to guess your final wish. Apropos, about the reason of the name RevolutionPlot3D[] we can something understand reading the page:

Cylindrical Coordinates