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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?

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  • $\begingroup$ Note that NIntegrate[1, {x, y} \[Element] Triangle[{{0, 0}, {1, 2}, {2, 1}}], Method -> {"MonteCarlo"}] returns unevaluated, too. $\endgroup$ – Michael E2 May 10 '15 at 21:45
  • $\begingroup$ I believe that mma regards {"MonteCarlo"} as "MonteCarlo",but it doesn't. $\endgroup$ – WateSoyan May 11 '15 at 1:12
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    $\begingroup$ I think they should be equivalent. I think there must be some sort of bug. When I try to Trace with TraceInternal -> True, it gobbles up memory and I have to kill it. $\endgroup$ – Michael E2 May 11 '15 at 1:40
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    $\begingroup$ Bug report filed, thanks for reporting this issue! $\endgroup$ – ilian May 13 '15 at 11:31
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    $\begingroup$ Fixed in version 10.2. $\endgroup$ – ilian Jul 14 '15 at 17:34
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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}]
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As mentioned in the comments, this bug has been fixed as of Mathematica 10.2.

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

(* 1.49258 *)
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