A visual tour via the Definite Integration $\int_{a}^{b}\frac{1}{x}dx$

TL;DR

This paper uses visual geometric interpretations of the integral of 1/x to illustrate classical mathematical results, including Euler's limit, inequalities, and constants, enhancing understanding through graphical methods.

Contribution

It introduces visual geometric representations for classical results like Euler's limit, inequalities, and constants, providing intuitive insights and alternative proofs.

Findings

Visual representation of Euler's limit and its generalization

Geometric proof of the inequality b^a < a^b for e ≤ a < b

Estimation of Euler's constant γ and the value of e near 2.7

Abstract

Geometrically, $\int_{a}^{b}\frac{1}{x}dx$ means the area under the curve $\frac{1}{x}$ from $a$ to $b$, where $0<a<b$, and this area gives a positive number. Using this area argument, in this expository note, we present some visual representations of some classical results. For examples, we demonstrate an area argument on a generalization of Euler's limit $\left(\lim\limits_{n\to\infty}\left(\frac{(n+1)}{n}\right)^{n}=e\right)$. Also, in this note, we provide an area argument of the inequality $b^a < a^b$, where $e \leq a< b$, as well as we provide a visual representation of an infinite geometric progression. Moreover, we prove that the Euler's constant $\gamma\in [\frac{1}{2}, 1)$ and the value of $e$ is near to $2.7$. Some parts of this expository article has been accepted for publication in Resonance - Journal of Science Education, The Mathematical Gazette, and International Journal of Mathematical Education in Science and Technology.