Browsing by Author "van Zyl, Christoff"

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    Implementation of the Cavalieri Integral
    (2023-02) van Zyl, Christoff; Grobler, Trienko; Stellenbosch University. Faculty of Science. Dept. of Computer Science.
    ENGLISH ABSTRACT: Cavalieri Integration in R n presents a novel visualization mechanism for weighted integration and challenges the notion of strictly rectangular integration strips. It does so by concealing the integrator inside the boundary curves of the integral. This paper investigates the Cavalieri integral as a superset of Riemann-integration in R n−1 , whereby the integral is defined by a translational region in R n−1 , which uniquely defines the integrand, integrator and integration region. In R 2 , this refined translational region definition allows for the visualization of Riemann-Stieltjes integrals along with other forms of weighted integration such as the Riemann–Liouville fractional integral and convolution operator. Programmatic implementation of such visualizations and computation of integral values are also investigated and relies on knowledge relating to numeric integration, algorithmic differentiation and numeric root finding. For the R 3 case, such visualizations over polygonal regions requires a mechanism for the triangulation of a set of nested polygons and transformations which allow for the use of repeated integration to solve the integration value over the produced triangular regions using standard 1-dimensional integration routines.