How to plot the ranges of the variables of an iterated integral?

Question

I would like to plot the volume of the following integral:

$$\int _{-1}^1\int _0^{\sqrt{1-y^2}}\int _{x^2+y^2}^{\sqrt{x^2+y^2}}\left(xyz\right)dzdxdy$$

Here are the ranges of the variables from this iterated integral.

$$-1\le y\le 1 $$

$$0 \le x\le \sqrt{1-y^2}$$

$$x^2+y^2\le z\le\sqrt{x^2+y^2}$$

This is the right half of a region which lower bound is an elliptic paraboloid and the upper bound is a cone.

I have no previous experience with Mathematica so I have only managed to plot a elliptic paraboloid and a cone:

ContourPlot3D[{z == x ^2 + y^2, z^2 == x ^2 + y^2}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]

I have tried this alternatives but the results are "empty":

RegionPlot3D[x^2 + y^2 <= (x^2 + y^2)^1/2, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}]

Plot3D[{ (x^2 + y^2), (x^2 - y^2)^1/2}, {x, -1, 1}, {y, -1, 1}, RegionFunction -> Function[{x, y}, (x^2 + y^2) < (x^2 - y^2)^1/2]]

RegionPlot3D[ z <= Sqrt[x^2 + y^2] && y^2 + x^2 < 0 && z > 0, {x, -1.5, 1.5}, {y, -2, 2}, {z, -2, 2}, PlotPoints -> 100]

Thanks in advance.

Answer 1

To get the region, you can use region plot, you should take the limits of the integral exactly as they are written and supply them to the function, separating the region corresponding to each integral by the And function(&&). The edge is a little jagged, but you can play with the number of plot points to get a better or worse picture.

RegionPlot3D[
 -1 <= y <= 1 && 0 < x < Sqrt[1 - y^2] && 
  x^2 + y^2 < z < Sqrt[x^2 + y^2]
 , {y, -1, 1}, {x, 0, 1}, {z, 0, 1}
 , PlotPoints -> 100
 , PerformanceGoal -> "Quality"
 , PlotStyle -> Directive[Yellow, Opacity[0.5]], Mesh -> None
 , BoxRatios -> {2, 1, 1}
 ]

Answer 2

Using ImplicitRegion

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

RegionPlot3D[rgn,
 PlotPoints -> 150,
 Axes -> True,
 AxesLabel ->
  (Style[#, Bold, 14] & /@ {"x", "y", "z"})]

The volume is

vol = Volume[rgn]

Pi/12

or alternatively,

vol == RegionMeasure[rgn, 3] ==
 Integrate[1, Element[{x, y, z}, rgn]] ==
 Integrate[1, {y, -1, 1}, {x, 0, Sqrt[1 - y^2]},
  {z, x^2 + y^2, Sqrt[x^2 + y^2]}]

True

Your original integral is

Integrate[x*y*z, Element[{x, y, z}, rgn]]

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