# How can I get a smooth plot of a bounded region?

From the iterated integrals $\int_{0}^{1}\int_{\sqrt{y}}^{1}\int_{x^{3}}^{1}f(x,y),$ we have the region $$\Omega=\{0\le y\le1,\sqrt{y}\le x \le 1,x^{3}\le z \le 1\}.$$

How can I use Mathematica to plot $\Omega$?

The following is what I tried.

RegionPlot3D[
x^3 <= z <= 1 && 0 <= y <= 1 && Sqrt[y] <= x <= 1,
{x, 0, 1}, {y, 0, 1}, {z, 0, 1},
PlotStyle -> Directive[Yellow, Opacity[0.5]],
Mesh -> None]


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Have you tried RegionPlot3D[]? – J. M. Feb 19 at 13:21

reg = ImplicitRegion[
0 <= y <= 1 && Sqrt[y] <= x <= 1 && x^3 <= z <= 1, {x, y, z}];
RegionPlot3D[reg, PlotPoints -> 100]


Don't forget the PlotPoints!

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RegionPlot3D will work fine, you just need to give it the proper region and specify the number of PlotPoints

RegionPlot3D[
ImplicitRegion[
x^3 <= z <= 1 && 0 <= y <= 1 && Sqrt[y] <= x <= 1, {x, y, z}],
PlotPoints -> 100, Axes -> True]


You can also use DiscretizeRegion

DiscretizeRegion[
ImplicitRegion[
x^3 <= z <= 1 && 0 <= y <= 1 && Sqrt[y] <= x <= 1, {x, y, z}]]


Edit ImplicitRegion is also very useful for integration.

Integrate[
Log[ x y], {x, y, z} ∈
ImplicitRegion[
x^3 <= z <= 1 && 0 <= y <= 1 && Sqrt[y] <= x <= 1, {x, y, z}]]
(* -(5/12) *)

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There is something error.Your picture is not the region $\Omega$. – AK47 Feb 19 at 13:42
@AK47 - sorry, I had copied and pasted the region from your original code. – Jason is no longer a postdoc Feb 19 at 13:46
This might be helpful ! – AK47 Feb 19 at 13:50