The aims for the work are to derive a numerical method to find a value for the double integral, which is integrable either continuously or continuously, but its derivative is singular or singular at one or both ends, for an integration using the author rule to find the error formulas associated with it, improving the results using Romberg’s acceleration. It delivers high-accuracy results quickly, even with a few partial periods. and comparing a result for a new rule and a result for rule (SS, TT, MM, ST, SM, MT) of accuracy and speed of approach.