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Monte Carlo integration, simple example and the results

N I \sigma_N
10 3.10263E+00 3.98802E-01
100 3.02933E+00 4.04822E-01
1000 3.13395E+00 4.22881E-01
10000 3.14195E+00 4.11195E-01
100000 3.14003E+00 4.14114E-01
1000000 3.14213E+00 4.13838E-01
10000000 3.14177E+00 4.13523E-01
10^{9} 3.14162E+00 4.13581E-01

We note that as N increases, the integral itself never reaches more than an agreement to the fourth or fifth digit. The variance also oscillates around its exact value 4.13581E-01 .