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

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 $\left(\underset{n\to \infty }{\lim }{\left(\frac{(n+1)}{n}\right)}^n=e\right)$. Also, in this note, we provide an area argument of the inequality $b^a<a^b$, where $e\le a<b$, 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.

arXiv comment10 pages, 15 figures