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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 positivenumber. Using this area argument, in this expository note, we present somevisual representations of some classical results. For examples, we demonstratean 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 aninfinite 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 partsof this expository article has been accepted for publication in Resonance -Journal of Science Education, The Mathematical Gazette, and InternationalJournal of Mathematical Education in Science and Technology.

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