[Paper Review] Riemann Integration in the Euclidean Space

Akerele Olofin Segun|arXiv (Cornell University)|Mar 21, 2024

Mathematics and Applications4 citations

TL;DR

This paper presents a rigorous extension of Riemann integration from one dimension to n-dimensional Euclidean space using Darboux upper and lower integrals. It establishes foundational concepts such as partitions, upper and lower sums, refinements, and integrability criteria, culminating in the equivalence of Darboux and Riemann integrals in ℝⁿ, with applications to evaluating definite integrals via Riemann sums.

ABSTRACT

The so-called Riemann sums have their origin in the efforts of Greek mathematicians to find the center of gravity or the volume of a solid body. These researches led to the method of exhaustion, discovered by Archimedes and described using modern ideas by MacLaurin in his extit{Treatise of Fluxions} in 1742. At this times the sums were only a practical method for computing an area under a curve, and the existence of this area was considered geometrically obvious. The method of exhaustion consists in almost covering the space enclosed by the curve with $n$ geometric objects with well-known areas such as rectangles or triangles, and finding the limit (though this topic was very blurry at these early times) when $n$ increases. One of its most remarkable application is squaring the area $\mathcal{A}$ enclosed by a parabola and a line.

Motivation & Objective

• To extend the classical Riemann integral from ℝ to ℝⁿ using the Darboux approach.
• To formalize the concepts of upper and lower Darboux integrals in higher dimensions.
• To establish the equivalence between Darboux and Riemann integrability in ℝⁿ.
• To demonstrate the utility of Riemann sums in evaluating definite integrals through concrete examples.
• To provide a systematic framework for understanding Riemann integration in multidimensional Euclidean spaces.

Proposed method

• Define partitions of bounded intervals in ℝⁿ as finite sets of points dividing the domain into subintervals.
• Introduce upper and lower Darboux sums using supremum and infimum of the function over subintervals.
• Establish the refinement property: refining a partition increases lower sums and decreases upper sums.
• Define the upper and lower Riemann integrals as the infimum of upper sums and supremum of lower sums over all partitions.
• Prove that the lower integral is always less than or equal to the upper integral for any bounded function.
• Use the squeeze theorem and telescoping series to evaluate limits of Riemann sums, connecting them to definite integrals.

Experimental results

Research questions

• RQ1 How can the Darboux approach to Riemann integration be generalized from ℝ to ℝⁿ?
• RQ2 What conditions ensure that the upper and lower Darboux integrals coincide in higher dimensions?
• RQ3 How do refinements of partitions affect the convergence of upper and lower sums?
• RQ4 In what ways can Riemann sums be used to evaluate definite integrals in multiple dimensions?
• RQ5 What is the relationship between the limit of Riemann sums and the value of a definite integral in ℝⁿ?

Key findings

• For any bounded function on a closed interval in ℝⁿ, the lower Darboux integral is always less than or equal to the upper Darboux integral.
• The Darboux integral exists if and only if the upper and lower integrals are equal, which is equivalent to Riemann integrability.
• The limit of Riemann sums for f(x) = x² over [2,5] converges to ∫₂⁵ x² dx = 39.
• The limit of the Riemann sum ∑(b−a)/n ⋅ (a + k(b−a)/n)⁻² as n→∞ equals ∫ₐᵇ x⁻² dx = 1/a − 1/b.
• The Bronstein integral ∫₀^π log(a² + b² − 2ab cos x) dx evaluates to 2π log(max{a,b}) when b > a.
• The evaluation of the Bronstein integral relies on complex factorization and the limit of a product of trigonometric terms, yielding π log(c²) where c = b/a > 1.