Informal discussion of the Euler's constant

I'm going to resort to the concept of the integral without the formalism. If you derive the natural log the result is the inverse of [math]\displaystyle{ x }[/math]. The other direction, if you calculate the area under the function's curve given by the inverse of [math]\displaystyle{ x }[/math] you get the natural log. The reason for this specific function [math]\displaystyle{ f(x) = 1/x }[/math] is that deriving logs in any base that isn't [math]\displaystyle{ e }[/math] yields [math]\displaystyle{ 1/xb }[/math], where [math]\displaystyle{ b \neq 1 }[/math] and is some log. With the natural log we have that [math]\displaystyle{ \ln(e) = 1 \iff e^1 = e }[/math], which implies that if we integrate [math]\displaystyle{ 1/x }[/math] from 1 to [math]\displaystyle{ e }[/math] the result is [math]\displaystyle{ e }[/math]. Why from 1? Because if we begin at [math]\displaystyle{ x = 0 }[/math] the function is undefined and beginning at 1 the first rectangle of the sum is a square of area equal to 1. For now it suffices to say that the concept of the integral that we first learn is a sum of rectangles under the function's curve. The base of the rectangle is [math]\displaystyle{ x_2 - x_1 }[/math] and the height is [math]\displaystyle{ f(x) }[/math] itself. You can even do it with a calculator and see that with more rectangles and shorter bases, the sum seems to approach [math]\displaystyle{ e }[/math].