# When to trust the result of numerical integration in mathematica?

How does one know when to trust the result of numerical integration, using NIntegrate, for higher (5 or 6) dimensional integrals in mathematica? For example, I get the following result

``````In(3):= Integrate( Exp(-a^2 - b^2 - c^2 - x^2 - y^2 - z^2), {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, {a, -2, 2}, {b, -2, 2}, {c, -2, 2}) // N

Out(3)= 30.1462
``````

whereas if I do the same integral numerically, I get

``````In(2):= NIntegrate( Exp(-a^2 - b^2 - c^2 - x^2 - y^2 - z^2), {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, {a, -2, 2}, {b, -2, 2}, {c, -2, 2}) // N

During evaluation of In(2):= NIntegrate::eincr: The global error of the strategy
GlobalAdaptive has increased more than 2000 times. The global error is expected
to decrease monotonically after a number of integrand evaluations. Suspect one
of the following: the working precision is insufficient for the specified
precision goal; the integrand is highly oscillatory or it is not a (piecewise)
smooth function; or the true value of the integral is 0. Increasing the value of
numerical integration. NIntegrate obtained 30.14615590437465 and
0.0001823733624988688 for the integral and error estimates.

Out(2)= 30.1462
``````

This seems to indicate that one can trust the result of NIntegrate if the error estimate is a small percentage (~1%) of the estimated value of the integral. If that’s true then how would one understand the discrepancy in the values obtained in the following two cases when Mathematica does not report any errors.

``````In(102):= NIntegrate( Exp(-a^2 - b^2 - c^2 - x^2 + y^2), {x, -2, 2}, {y, -2, 2}, {a, -2, 2}, {b, -2, 2}, {c, -2, 2}, Method -> "AdaptiveMonteCarlo") // N

Out(102)= 298.918

In(104):= NIntegrate( Exp(-a^2 - b^2 - c^2 - x^2 + y^2), {x, -2, 2}, {y, -2, 2}, {a, -2, 2}, {b, -2, 2}, {c, -2, 2}, Method -> "MonteCarloRule") // N

Out(104)= 313.592
``````

In the following example, I also get a result with an error bar that is a small (~2%) percentage of the value of the integral

``````In(7):= NIntegrate( Exp(-a^2 - b^2 - c^2 - x^2 - y^2 - z^2), {x, -5, 5}, {y, -5, 5}, {z, -5, 5}, {a, -5, 5}, {b, -5, 5}, {c, -5, 5}, Method -> "AdaptiveQuasiMonteCarlo") // N

During evaluation of In(7):= NIntegrate::maxp: The integral failed to converge
after 1000100 integrand evaluations. NIntegrate obtained 0.29686054547957375 and
0.005304460608762476 for the integral and error estimates.

Out(7)= 0.296861
``````

but comparing it with the following analytic evaluation of the integral

``````
Out(8)= 31.0063
``````

we see that the answer is way off. Any guidance in this regard will be greatly appreciated.