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]

But the edge is bad.

Replay

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!

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) *)

Category: plotting Time: 2016-07-30 Views: 11