# ImplicitRegion with RegionPlot Example

Using Mathematica 11.0.1.0 on a MacBook Pro (OSX 10.11.6) I tried this:

Clear[x, y, reg1]
reg1 = ImplicitRegion[
Log[10, 1 + x^2 + y^2] <= 1 + Log[10, x + y], {x, y}];
RegionPlot[reg1]


But got only this output:

RegionPlot[reg1]

But I did get the area.

Area[reg1]


Which gave the correct answer $49\pi$. Then I tried:

RegionPlot[
Log[10, 1 + x^2 + y^2] <= 1 + Log[10, x + y], {x, -2, 12}, {y, -2,
12}]


And I did get the correct image:

Is the fact that RegionPlot[reg1] did not work a bug that should be reported, or have I made some sort of mistake?

• RegionPlot[region] and RegionPlot[inequality, vars...] are not equivalent. For instance, try DiscretizeRegion[reg1] and DiscretizeRegion[reg1, Method -> "RegionPlot"]. I think your (failed) RegionPlot uses the first approach. Commented Dec 3, 2016 at 22:00

THIS IS JUST AN EXTENDED COMMENT

The region definition can be simplified to

reg2 = ImplicitRegion[
1 + x^2 + y^2 <= 10 (x + y),
{x, y}];

Area[reg2]

(*  49 π  *)

Show[
RegionPlot[reg2],
ContourPlot[
Log10[1 + x^2 + y^2] == 1 + Log10[x + y],
{x, -2, 12}, {y, -2, 12},
ContourStyle -> Red,
PlotPoints -> 50]]