Interpolatory quadrature rules over general non-rectangular domains

This paper presents a unified framework for constructing interpolatory quadrature rules on general non-rectangular domains whose integration limits depend on preceding coordinates. Such domains arise naturally in hierarchical integration, finite element analysis, and geophysical modeling, yet a general and systematic quadrature theory has been lacking. The proposed approach is based on a recursive extension of multidimensional Lagrange interpolation, which directly incorporates the variable-dependent structure of the integration region. Within this framework, we derive explicit Newton–Cotes and Gauss–Legendre type quadrature rules in arbitrary dimensions, together with corresponding remainder expressions. A key feature of the method is that quadrature rules are constructed directly on the physical domain without introducing mapping-induced geometric distortion. This allows a clear distinction between direct-domain quadrature and Jacobian-based mapping approaches. In particular, equivalence with classical mapped formulations is recovered for affine geometries, while for curved domains the proposed method avoids projection-induced errors inherent in reference-domain transformations. The resulting framework provides a consistent and flexible approach for high-order numerical integration over complex domains and offers a foundation for further developments in multidimensional quadrature theory.