Visualization of Fractional Integrals

TL;DR

This paper introduces a new geometric interpretation of the Riemann-Liouville fractional integral, visualizing it as an infinite sum of non-rectangular infinitesimal areas, enhancing pedagogical understanding.

Contribution

It provides the first geometric interpretation of fractional integrals, linking them to familiar Riemann integral concepts for improved teaching and comprehension.

Findings

Fractional integral visualized as an infinite sum of infinitesimal areas

Geometric interpretation similar to Riemann integral

Pedagogical benefits for teaching fractional calculus

Abstract

We presented a novel geometric interpretation of the Riemann-Liouville fractional integral. We found that a Riemann-Liouville integral can be thought of as the area obtained by summing together the area of an infinite number of non-rectangular infinitesimals whose shape is determined by the order of integration {\alpha} and the integration limit t. We also showed that this geometric interpretation offers many pedagogical benefits as it is very similar in nature to the geometric interpretation of the Riemann integral.