The primary region. The first rule is composed of Boole’s rule for the inner dimension and the midpoint rule for the outer dimension. The second rule is composed of Boole’s rule on both internal dimensions. The goal of the research is to formulate two rules for determining the values of the continuous double integral inside the integration x, y, with the extraction of the correction limits that come with the two rules, and making use of them by extracting them, using the Romberg-Regardson acceleration, and comparing the accuracy and speed of approach to the exact value.