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This is perhaps too "creative". Some health checks needed for the series behavior:

Graphics`Region`RegionInit[];
region = (x x + 4 y y + 9 z z <= 1);
paregion = Region`ParametricRegion[{{x, y, z}, region}];
k = FullSimplify@
     Normal@Series[Sqrt[(1-9 z^2) (1-9 z^2-4 y^2)], {z,0, #},{y,0, #}] &/@ Range[1, 10, 2];
res = N@Integrate[#, {x, y, z} ∈ paregion] & /@ k

(* {0.698132, 0.488692, 0.480154, 0.477423, 0.476201} *)

So the result is near to 0.476

ListLinePlot@res