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Density plot roh(x,y,z)=1+x+y+z bounded by x^2+y^2=1 and y=z, x=0, z=0. In the first octant

So far,

roh[x_, y_, z_] := 1 + x + y + z;

DensityPlot3D[roh[x, y, z], {x, 0, 1}, {y, 0, x^2}, {z, 0, 1}]

Then I need a point at the center of mass. I think perhaps my region is wrong but nothing I try seems to work. I have tried {y,0, Sqrt[1-x^2]} and a few others.

For the center of mass I tried

McX = NIntegrate[y*roh[x, y,z], {x, 0, 2}, {y, 0, },{}];
McY = NIntegrate[x*roh[x, y,z], {x, 0, 2}, {y, 0, },{}];
{X, Y} = {McY/M, McX/M};

z is blank because I was just trying different things

then to plot the point Graphics[{PointSize[.04], Black, Point[{X, Y}]}]

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1 Answer 1

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Clear["Global`*"]

roh[x_, y_, z_] = 1 + x + y + z;

rgn = ImplicitRegion[
   x^2 + y^2 <= 1 && y > z >= 0 && x >= 0, {x, y, z}];

DensityPlot3D[roh[x, y, z],
 {x, y, z} ∈ rgn,
 PlotPoints -> 100,
 PerformanceGoal -> "Quality",
 AxesLabel -> (Style[#, 12, Bold] & /@ {x, y, z}),
 PlotLegends -> BarLegend[Automatic, LegendLabel -> roh]]

enter image description here

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