Unable to compute the measure of a region in four dimensions

Question

I need to be able to calculate an integral with this implicit region:

ImplicitRegion[
  16 π^2 - 
    Sqrt[32 π^2 + Abs[Cos[θ1] r1]^2 + 
      Abs[r1 Sin[θ1]]^2] Sqrt[
     Abs[Cos[θ1] r1]^2 + 
      Abs[Cos[φ1] r1 Sin[θ1]]^2 + 
      Abs[r1 Sin[φ1] Sin[θ1]]^2] + Cos[θ1]^2 r1^2 +
     Cos[φ1] r1^2 Sin[θ1]^2 <= 0.5 && 
  16 π^2 - 
    Sqrt[32 π^2 + Abs[Cos[θ1] r1]^2 + 
      Abs[r1 Sin[θ1]]^2] Sqrt[
     Abs[Cos[θ1] r1]^2 + 
      Abs[Cos[φ1] r1 Sin[θ1]]^2 + 
      Abs[r1 Sin[φ1] Sin[θ1]]^2] + Cos[θ1]^2 r1^2 +
     Cos[φ1] r1^2 Sin[θ1]^2 >= -0.5 && 
  16 π^2 - 
    Abs[r2] Sqrt[32 π^2 + Abs[r2]^2 + 
        Abs[r1 Sin[φ1] Sin[θ1]]^2 + 
        Abs[r1 Sin[θ1] - Cos[φ1] r1 Sin[θ1]]^2] + 
    r2^2 <= 0.5 && 
  16 π^2 - 
    Abs[r2] Sqrt[32 π^2 + Abs[r2]^2 + 
        Abs[r1 Sin[φ1] Sin[θ1]]^2 + 
        Abs[r1 Sin[θ1] - Cos[φ1] r1 Sin[θ1]]^2] + 
    r2^2 >= -0.5 && 160 <= r1 <= 300 && 160 <= r2 <= 300 && 
  0 <= φ1 <= 2 π && 0 <= θ1 <= 2 π, 
  {r1, r2, φ1, θ1}]

I'm sorry if it's difficult to visualize, but I wouldn't know how to do it properly in a reasonable time here. Anyway, it's the same expression four times basically ( $f(\theta1,\phi1, r1,r2) \leq 0.5)$ and Mathematica tells me:

Unable to compute the dimension of region

and I don't understand what's wrong, apart from the fact that it might have problems because it's in 4D. How could I formulate it differently?

Edit

I managed to simplify it by making some assumptions:

ImplicitRegion[
  -633.655 <= 
  r1 (r1 (4. + 1. θ1^4 - 2. θ1^2 ϕ1^2) - 
        Sqrt[(128 π^2 + r1^2 (4 + θ1^4)) (4 + θ1^4 + θ1^2 ϕ1^4)]) <= 
  -629.655 && 
  314.827 + 2. r2^2 <= 
  r2 Sqrt[128 π^2 + 4 r2^2 + r1^2 θ1^2 ϕ1^2 (4 + ϕ1^2)] && 
  r2 Sqrt[128 π^2 + 4 r2^2 + r1^2 θ1^2 ϕ1^2 (4 + ϕ1^2)] <= 
  316.827 + 2 r2^2 &&
  r1 >= 0 && r2 >= 0 && 160 <= r1 <= 200 && 160 <= r2 <= 200 &&
  -0.1 <= θ1 <= 0.1 && -0.1 <= ϕ1 <= 0.1, 
 {r1, r2, θ1, ϕ1}]

Now it doesn't give that error anymore, but instead it never finishes calculating.

Edit

I need to integrate a function over that region: NIntegrate[fn[r1, r2, θ1, ϕ1], {r1, r2, ϕ1, θ1} ∈ IntegrationZone, Method -> "MonteCarlo"] but what my problem is, is that NIntegrate[], or more precisely, Volume[], cannot calculate the area of that function. If I change the method to e.g. GlobalAdaptive, I get:

iCopyExpr() called on symbol.

so that's why I'm trying MonteCarlo.

Answer 1

You can use Boole and integrate numerically:

NIntegrate[
Boole[-633.655`30 <= r1 (r1 (4+ θ1^4-2 θ1^2 ϕ1^2) - Sqrt[(128 π^2+r1^2 (4+θ1^4)) (4+θ1^4+θ1^2 ϕ1^4)]) <= -629.655`30 && 314.827`30+2 r2^2 <= r2 Sqrt[128 π^2+4 r2^2+r1^2 θ1^2 ϕ1^2 (4+ϕ1^2)] && r2 Sqrt[128 π^2+4 r2^2+r1^2 θ1^2 ϕ1^2 (4+ϕ1^2)] <= 316.827`30+2 r2^2]
, {r1,160,200}, {r2,160,200}, {θ1,-1/10,1/10}, {ϕ1,-1/10,1/10}
, Method->"MonteCarlo", WorkingPrecision->30]

I get the result $61.4(5)$. You can probably reduce WorkingPrecision to machine precision and get the same result; I leave this to you.