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Need help with ideas to make NIntegrate a little faster for multiple variables

I'm finding probabilities of a Gaussian ball that has been
sliced by some hyperplanes.  Calculations are taking a LONG time.

Would anyone have any advice on how I might find solutions
to problems like the following in a more reasonable amount
of time that it seems to take in the current form?


The surface is about as smooth as could be expected for a problem.

The real bounds of integration are infinity but I would be able to use
results that have 4 good digits in an answer and thus I have been using
bounds of +/-5 or +/-10 instead of Infinity.  But that doesn't seem to
be able to give results in a few hours (on a 360 Mhz machine with lots
of free memory and nothing else soaking up cycles)

I have tried setting NumPoints->10000000 which gives perhaps 10 points
per dimension plus some spares for the algorithm to use as it sees fit.
But Version 3 claims that it sees an error when NumPoints is set and thus
reverts to not using the compiled version I think.  The calculations
continue, but slowly, in either case.

Does anyone have any ideas about what I might do to get answers with a
handful of good digits in a plausible amount of time?  I believe I have
reduced the problem down to the simplest possible form.  But I am still
left with sixty four such integrals to evaluate, to at least give me
eight good points to plot on a graph.

(Would upgrading to version 4 make a dramatic change in this particular

Many Thanks

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