[?] Calculate the volume of an exponentially distorted ellipsoid?

Questions
,
1

Consider the following code:

a = 2; b = 1.5; c = 1; tx = 0.175; ty = 0.2; tz = 0.25;
    AbsoluteTiming[
     Volume[ImplicitRegion[
       Exp[tx*x] (x/a)^2 + Exp[ty*y] (y/b)^2 + Exp[tz*z] (z/c)^2 <= 1, {x,
         y, z}], Method -> "NIntegrate"]]
    
 {39.1684, 
 RegionMeasure[
  ImplicitRegion[
   1/4 E^(0.175 x) x^2 + 0.444444 E^(0.2 y) y^2 + E^(0.25 z) z^2 <= 
    1, {x, y, z}], 3, WorkingPrecision -> MachinePrecision, 
  Method -> "NIntegrate"]}
2

Alternative way:

 $Version
 (* "11.3.0 for Microsoft Windows (64-bit) (March 7, 2018)" *)

 a = 2; b = 1.5; c = 1; tx = 0.175; ty = 0.2; tz = 0.25;
 R = ImplicitRegion[Exp[tx*x] (x/a)^2 + Exp[ty*y] (y/b)^2 + Exp[tz*z] (z/c)^2 <= 1, {x,y, z}];
 V = DiscretizeRegion[R, Method -> "RegionPlot3D"] // Quiet;
 Volume[V]

(* 6.924361674 *)

or:

a = 2; b = 1.5; c = 1; tx = 0.175; ty = 0.2; tz = 0.25;
AbsoluteTiming[N[Volume[ImplicitRegion[Exp[tx*x]*(x/a)^2 + Exp[ty*y]*(y/b)^2 + Exp[tz*z]*(z/c)^2 <= 1, {x, y, z}]]]]

(*  {202.8073994, 6.924361674} *)
3

Thanks

4

I had to correct the answer because I had a small mistake.

Regards,MI