Hi,
I've not looked in detail, but for your calculation, it looks to me that 
you don't need an integral quantity.
Just use a postprocessing : 
      { Name Tave; Value { Integral { [ what you want ] ; In Sfc ;
                   Integration I ; Jacobian J ; }}}
and then in PostOperation :
      Print[ Tave [Sfc], OnGlobal , File "surf.result", Format Table];
(tell me if I'm wrong)
Regards,
O.C.