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I have an issue : 

NIntegrate[x^2 *Exp[-x^2], {x, 0, Infinity}]

gives out 0.443113

But :

NIntegrate[x^2 *Exp[-x^2], {x, 0, 6857}]

gives an error:

NIntegrate::izero: Integral and error estimates are 0 on all integration subregions. Try increasing the value of the MinRecursion option. If value of integral may be 0, specify a finite value for the AccuracyGoal option.

and a zero as a result

However:

NIntegrate[x^2 *Exp[-x^2], {x, 0, 6855}]

gives the correct answer of 0.443113.

What is the difference of integrating over 6855 or 6857 ? (I am not really aware of the insides of mathematica and how it calculates the integrals numerically).

Addition to my question - how to access the properties of integration and play with the parameters.

Thank you a lot for your help :)

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  • $\begingroup$ Note that NIntegrate's message "izero" pretty much tells you how to get a better result. As for "how to access the properties of integration and play with the parameters." -- all of these are answered in the advanced NIntegrate documentation. $\endgroup$
    – Anton Antonov
    Jan 28, 2019 at 15:42
  • $\begingroup$ What version have you observed this with? I cannot reproduce your results and message in 11.3: for both sets of limits I get 0.443113 . $\endgroup$
    – Anton Antonov
    Jan 28, 2019 at 18:56
  • $\begingroup$ @AntonAntonov One guess is that in V11.3, machine underflow (General::munfl) is used as a signal for recursive subdivision. Certainly a Trace[] indicates that the underflow message is suppressed. $\endgroup$
    – Michael E2
    Jan 29, 2019 at 3:48
  • $\begingroup$ @AntonAntonov On 11.3, I had to go a little higher than 6857 to induce the problem, but I didn't think it was important enough to highlight in my answer. $\endgroup$
    – John Doty
    Jan 29, 2019 at 12:31

1 Answer 1

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x^2*Exp[-x^2] is effectively 0 at most points in your interval.

Reduce[x^2*Exp[-x^2] >= $MinMachineNumber]
(* -26.7389 <= x <= -1.49167*10^-154 || 1.49167*10^-154 <= x <= 26.7389 *)

NIntegrate works by sampling the function. If it never finds a nonzero sample, you'll see the error. But the message gives you the advice to fix the problem.

NIntegrate[x^2*Exp[-x^2], {x, 0, 10000}, MinRecursion -> 5]
(* 0.443113 *)
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