relation between integral and summation What is the relation between a summation and an integral ? This question is actually based on a previous question of mine here  where I got two answers (one is based on summation notation) and the other is based on integral notation and I do not know yet which one to accept . So I would like to understand the connect between the two ?

[The Summation notation was solved using the logic that the area under a function $f(x)$ is the sum of the rectangles with very very small width. 
In this equation the $b/n$ and $a/n$ is the given width, where $n$ approaches infinity ($n$ is the number of rectangles you want it to be divided into). 
The length of the rectangle is the value of the function at a given $x$. To use the summation notation $b/n$ and $a/n$ is multiplied to $i$ so as to get the length (value of $f(x)$) as it moves along the function towards the limit ($b$ or $a$) and $i$ approaches $n$.
To calculate the area, the summations for $b$ and $a$ were just simply subtracted with both have reference point from zero.
$$\int_a^b f(x)dx = \frac{b}{n} \sum\limits_{i=1}^n f\left(\frac{bi}{n}\right) - \frac{a}{n} \sum\limits_{i=1}^n f\left(\frac{ai}{n}\right)\text{ where $n \to \infty$}$$
