Get right results using RegionPlot3D command in Spherical Coordinates?

Questions
1

Hi;

In using the RegionPlot3D command to print a hemisphere of radius 3 using spherical coordinates, I end up with something that looks sort-of like a rectangle (see attached). It seems that if you use rectangular coordinates you can create a hemisphere using the RegionPlot3D command, but it certainly doesn’t seem to like spherical coordinates. Do I need to do something special when entering ranges in spherical coordinates into the RegionPlot3D command?

Thanks,

2

I would use ParametricRegion. In line of principle, I would expect this to give a hemisphere:

DiscretizeRegion@ParametricRegion[
  {FromSphericalCoordinates[{\[Rho], \[Theta], \[Phi]}],
   0 <= \[Rho] <= 3 && 0 <= \[Theta] <= 2 \[Pi] && 
    0 <= \[Phi] <= Pi/2},
  {\[Rho], \[Theta], \[Phi]}]

Instead, I get a cube. The two dimensional analogue works fine:

DiscretizeRegion@
 ParametricRegion[{FromPolarCoordinates[{\[Rho], \[Theta]}],
   0 <= \[Rho] <= 3 && 0 <= \[Theta] <= 2 \[Pi]},
  {\[Rho], \[Theta]}]