Volume integration with a condition
Clear["Global`*"]
Using If
vol1 = NIntegrate[
If[ArcSin[x1] + ArcSin[x2] + ArcSin[x3] + ArcSin[x4] > 3 Pi/2, 1, 0], {x4,
0, 1}, {x3, 0, 1}, {x2, 0, 1}, {x1, 0, 1}, MinRecursion -> 9,
WorkingPrecision -> 20,
Method -> {"AdaptiveQuasiMonteCarlo", "RandomSeed" -> 1234}]
(* 0.00076851819200000000000 *)
Using Boole
vol2 = NIntegrate[
Boole[ArcSin[x1] + ArcSin[x2] + ArcSin[x3] + ArcSin[x4] > 3 Pi/2], {x4, 0,
1}, {x3, 0, 1}, {x2, 0, 1}, {x1, 0, 1}, MinRecursion -> 9,
WorkingPrecision -> 20,
Method -> {"AdaptiveQuasiMonteCarlo", "RandomSeed" -> 1234}]
(* 0.00076851819200000000000 *)
Using Piecewise
vol3 = NIntegrate[
Piecewise[{{1,
ArcSin[x1] + ArcSin[x2] + ArcSin[x3] + ArcSin[x4] > 3 Pi/2}}], {x4, 0,
1}, {x3, 0, 1}, {x2, 0, 1}, {x1, 0, 1}, MinRecursion -> 9,
WorkingPrecision -> 20,
Method -> {"AdaptiveQuasiMonteCarlo", "RandomSeed" -> 1234}]
(* 0.00076851819200000000000 *)
vol1 == vol2 == vol3
(* True *)