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

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.