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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]

But the edge is bad.

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2 Answers 2

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

enter image description here

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]

enter image description here

You can also use DiscretizeRegion

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

enter image description here

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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  • $\begingroup$ There is something error.Your picture is not the region $\Omega$. $\endgroup$
    – AplehKevin
    Feb 19, 2016 at 13:42
  • $\begingroup$ @AK47 - sorry, I had copied and pasted the region from your original code. $\endgroup$
    – Jason B.
    Feb 19, 2016 at 13:46
  • $\begingroup$ This might be helpful ! $\endgroup$
    – AplehKevin
    Feb 19, 2016 at 13:50

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