Three rules are derived for numerically determining the values of double integrals utilizing the middle point and three-eighths criteria

Abstract

By utilizing Romberg accelaration and the twonumericals rules( the middle point and the three-eghits) and their corresponding error formulas, we hope to derive three numerical rules that can be used to extract the values of continuous binary integrals during the integration period. Applying the three principles produced results that were accurate to fourteen or fiteen decimal places and had comparatively  few subintervals on the integral's inner and outer dimensions inside the region of integration.