To find the volume of the following region:

fn[x_, y_, z_]:= Abs[0.7*x*Exp[I*y] + 0.3*Sqrt[x^2 + 8*10^-5]   + Sqrt[x^2 + 3*10^-3]*0.02*Exp[I*z]]  R = ImplicitRegion[fn[x, y, z]<=3*10^-3, {{x, 0, 0.015}, {y, 0, 2*Pi}, {z, 0, 2*Pi}}]  RegionPlot3D[  fn[x, y, z] <= 3*10^-3, {x, 0, 0.015}, {y, 0, 2*Pi}, {z, 0, 2*Pi},   PlotPoints -> 50, AxesLabel -> {x, y, z},   PlotStyle -> Directive[Yellow, Opacity[0.5]], Mesh -> None] 

Use “Volume”:

Volume[R] 

Volume::nmet: Unable to compute the volume of region

Use “NIntegrate”:

NIntegrate[   Boole[fn[x, y, z] <= 3*10^-3], {x, 0, 0.015}, {y, 0, 2*Pi}, {z, 0,     2*Pi}, Method -> {"GlobalAdaptive", "MaxErrorIncreases" -> 50000},    PrecisionGoal -> 3] // Timing 

NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.

NIntegrate::eincr: The global error of the strategy GlobalAdaptive has increased more than 50000 times. The global error is expected to decrease monotonically after a number of integrand evaluations… NIntegrate obtained 0.12219642052793653 and 0.007515502934285486 for the integral and error estimates.

{19.25, 0.122196}

The result is not convergent.

Use “MonteCarlo”:

NIntegrate[   Boole[fn[x, y, z] <= 3*10^-3], {x, 0, 0.015}, {y, 0, 2*Pi}, {z, 0,     2*Pi}, Method -> {"MonteCarlo", "MaxPoints" -> 10^12,      "RandomSeed" -> 9}, PrecisionGoal -> 3] // Timing 

{8.39063, 0.121479}

It is slow, and if set PrecisionGoal -> 5,

NIntegrate[  Boole[fn[x, y, z] <= 3*10^-3], {x, 0, 0.015}, {y, 0, 2*Pi}, {z, 0,    2*Pi}, Method -> {"MonteCarlo", "MaxPoints" -> 10^12,     "RandomSeed" -> 9}, PrecisionGoal -> 5] 

Running more than 2 minutes and can’t give a result.

Hope to find a better way to calculate the volume of this region, and realize:

1.high accuracy(at least PrecisionGoal -> 5)

2.good error estimate

3.fast

Thanks for your time.