This integral
i1 = NIntegrate[ Sqrt[1 + (x^6 + y^6)/(1 - x^4 - y^4)^(3/2)] ,
{x, 0, 1/3^(1/4)}, {y, 0, x}, WorkingPrecision -> 100, PrecisionGoal -> 40 ]
returns, roughly, 0.30155… along with the warning
“NIntegrate obtained 0.30155… and 3.20396…*10^-16 for the integral and error estimates”
If the estimate 3.20396…*10^-16 is an upper bound, this is valuable information.
But if the actual error could be larger than this, then this number is not useful (to me).
Which is it? A mere estimate of the error, or an upper bound on the error?