numerical integration - Why does Nintegrate keep unevaluated?

Bug introduced in 10.0 and fixed in 10.2.0

It's no surprise that the "MonteCarlo" Method works well:

NIntegrate[1, {x, y} ∈ Triangle[{{0, 0}, {1, 2}, {2, 1}}], 

 Method -> "MonteCarlo"]
(*1.50189*)

But when I try to change "MaxPoints", NIntegrate directly refuses me.

NIntegrate[1, {x, y} ∈ Triangle[{{0, 0}, {1, 2}, {2, 1}}], 
 Method -> {"MonteCarlo", "MaxPoints" -> 10^5}]

So sad! What's the story?

Answer

I wonder if this is a bug that appears when using regions and {"MonteCarlo"} as a method. It hangs my machine. This might be a possible workaround:

NIntegrate[1, {x, y} \[Element] Triangle[{{0,0},{1,2},{2,1}}],
 Method-> "MonteCarlo", MaxPoints -> 10^5}]

There isn't mention of the new arbitrary region functionality with the "MaxPoints" option or "MonteCarloRule" option in the documentation. You could also convert an arbitrary region with Boole[RegionMember[...]] and specify some ranges for x and y. This works on my machine:

NIntegrate[Boole[RegionMember[
  Triangle[{{0, 0}, {1, 2}, {2, 1}}],{x, y}]],
 {x,-10,10}, {y,-10,10}, Method -> {"MonteCarlo", "MaxPoints" -> 10^5}]

Comments

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