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Difference Integration is the same as Euler Integration except in the recording of the results.  In Euler Integration, the results are recorded at the end of step 3 above.  With difference Integration the results are recorded at the beginning of Step 3, before the new rates have been computed.  Put another way, Euler integration reports Levels and the values that result from those Levels, whereas difference integration reports the Level and the values that resulted in those Levels.