ROOT offers two paths for multidimensional integration.
The first is ROOT::Math::AdaptiveIntegratorMultiDim, the second is ROOT::Math::GSLMCIntegrator.
ROOT::Math::AdaptiveIntegratorMultiDim asks for a relative tolerance, an absolute tolerance and a maximum number of points. An minimum number of points is also specifiable. The documentation fails to make entirely clear that it ignores the absolute tolerance.
ROOT::Math::GSLMCIntegrator one-ups this. It requests absolute tolerance, relative tolerance and maximum number of calls, but apparently ignores both tolerances and uses the maximum number of calls each time. This again, is not identifiably documented.
These both share an interface through ROOT::Math::IntegratorMultiDim, which is where the practical value of the arguments should probably be made most clear. I understand the need for extensible interfaces, but when you are ignoring crucial sounding arguments, it is probably prudent to inform the user.