论文标题
通过确定集成的视觉游览$ \ int_ {a}^{b} \ frac {1} {x} dx $
A visual tour via the Definite Integration $\int_{a}^{b}\frac{1}{x}dx$
论文作者
论文摘要
几何，$ \ int_ {a}^{b} \ frac {1} {x} dx $表示曲线$ \ frac {1} {x} {x} $从$ a $ a $ a $ b $，其中$ 0 <a <a <b $，并且该区域给出了正数。在此说明中，使用此区域参数，我们提供了一些经典结果的一些视觉表示。例如，我们演示了一个区域参数，以了解Euler的限制$ \ left（\ lim \ limits_ {n \ to \ infty} \ left（\ frac {（n+1）} {n}} {n} \ right）^{n} {n} = e \ right）$。另外，在本说明中，我们提供了不平等$ b^a <a^b $的领域论点，其中$ e \ e \ leq a <b $，以及我们提供了无限几何进步的视觉表示。此外，我们证明了Euler在[\ frac {1} {2}，1）$中的常数$γ\，$ e $的值接近$ 2.7 $。这本说明性文章的某些部分已被接受用于共鸣的出版 - 《科学教育杂志》，《数学公报》和《国际科学与技术数学教育杂志》。
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 $γ\in [\frac{1}{2}, 1)$ 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.
