Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Riemann Integration in the Euclidean Space

Akerele Olofin Segun

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 \textit{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.

Riemann Integration in the Euclidean Space

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 \textit{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 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 increases. One of its most remarkable application is squaring the area enclosed by a parabola and a line.
Paper Structure (19 sections, 15 theorems, 144 equations)

This paper contains 19 sections, 15 theorems, 144 equations.

Table of Contents

  1. The Riemann Integral in $\mathbb{R}$
  2. Definition.
  3. Remark.
  4. Definition.
  5. Remark.
  6. Riemann Integral Over Rectangles
  7. Rectangles and Partitions
  8. Upper and Lower Integrals
  9. The Riemann Integral in $\mathbb{R}^n$
  10. Integrals of Continuous functions
  11. Integration of functions with Compact Support
  12. Iterated Integrals and Fubini's Theorem
  13. Example
  14. Examples.
  15. The Set of Riemann Integrable Functions
  16. ...and 4 more sections

Key Result

Lemma 1.1

Let $f$ be a bounded function on $[a,b]$. If $m\leq f(x)\leq M$ for all $x\in[a,b]$, then for any partition $\mathcal{P}$ of $[a,b]$

Theorems & Definitions (38)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Theorem 1.3
  • proof
  • Proposition 1
  • proof
  • Definition 1
  • Definition 2
  • ...and 28 more