A Jacobian-Based Generalization of Volume of a Solid of Revolution Along an Oblique Axis

Traditional calculus methods for solids of revolution are restricted to vertical or horizontal axes, leaving a gap in analytical rigor and practical application when dealing with oblique rotations. This study aimed to formalize the computation of volumes of solids of revolution along oblique axes by introducing Jacobian-based scaling factors that validate earlier geometric approaches and extend their utility to construction engineering. Using a generalized Jacobian algorithm, distinct factors were derived for the Disk Method (longitudinal scaling) and Shell Method (radial orientation), and applied to a benchmark problem involving parabolic boundaries rotated about oblique lines. Results confirmed that the Disk Jacobian functions as a slope factor correcting for axis stretch, while the Shell Jacobian ensures layering accuracy in curved structures, with both methods converging to identical outcomes and guaranteeing 100% volumetric consistency. Engineering applications demonstrated improved precision in material estimation for slanted members and complex forms, reducing resource waste and supporting sustainable practices. By bridging theoretical calculus with applied engineering, the framework advances SDG 9 (Industry, Innovation, and Infrastructure) through mathematical innovation and SDG 12 (Responsible Consumption and Production) by optimizing resource efficiency. This analytical contribution elevates oblique-axis rotations from geometric intuition to a reliable computational standard, offering professionals a generalizable tool for sustainable infrastructure design.