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

. Geometrically, (cid:82) b a 1 x dx means the area under the curve 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 (cid:16) lim n →∞ (cid:16) ( n +1) n (cid:17) n = e (cid:17) . Also, in this note, we provide an area argument of the inequality b a < a b , where e ≤ a < b , as well as we provide a visual representation of an infinite geometric progression. Moreover, we prove that the Euler’s constant γ ∈ [ 12 , 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.