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 *)