The Theoretical Basis of Double Integration

Abstract

Mathematical knowledge is divided into theoretical conceptual knowledge and procedural applied knowledge, and the balance between the two forms a fundamental condition for developing deep mathematical understanding. Double integration is considered one of the central concepts in advanced mathematics. However, studies often focus on the procedural and applied aspects while neglecting the theoretical aspect. This paper aims to present the theoretical foundations of double integration as an extension of single-variable definite integration. The descriptive-analytical method was used to study mathematical concepts and relationships. The research addresses the definition of double integration through the double Riemann sum and its existence conditions and simplifies its calculation according to Fubini's theorem and coordinate transformation, starting with its study over rectangular regions and then generalizing to general regions in the plane via the extension of the integrable function. The research shows that procedural knowledge of double integration cannot be complete without understanding the theoretical foundations that enable the researcher to accurately represent natural phenomena and translate them into mathematical models, thereby highlighting the importance of balancing the theoretical and applied aspects.