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Strange, I don't seem to have trouble with
sage: integral(integral(e^(-0.00260657639223762*(h - 11.1600000000000)^2 - 1.34372480515*(d - 2.85000000000000)^2)/pi, d, 0,Infinity), h, 0, Infinity).n()
13.3454645567847
or
sage: f(d,h) = (1/(2*pi*0.61*13.85))*exp(-1/2*((d-2.85)^2/0.61^2+(h-13.85)^2/13.85^2))
sage: integral(integral(f(d,h), d, 0,Infinity), h, 0, Infinity).n()
0.841343497874347
however, the answer to problems with symbolic integration and then numerical evaluating suggests using .n(prec) or RealIntervalField(prec) so, for example, you could try
sage: integral(integral(e^(-0.00260657639223762*(h - 11.1600000000000)^2 - 1.34372480515*(d - 2.85000000000000)^2)/pi, d, 0,Infinity), h, 0, Infinity).n(prec=12)
13.3
sage: integral(integral(f(d,h), d, 0,Infinity), h, 0, Infinity).n(prec=12)
0.842
sage: RealIntervalField(12)(integral(integral(f(d,h), d, 0,Infinity), h, 0, Infinity).n())
0.842?
sage: RealIntervalField(30)(integral(integral(f(d,h), d, 0,Infinity), h, 0, Infinity).n())
0.841343498?
(does the RealIntervalField(12) answer indicate an error in RealIntervalField() !?)
