What is the difference between the integral$\int\limits_{}^{}$ and segments $\sum_{} ^{} $ I want to know, what is the difference between the integral and segments
I know that، Integration came when scientists asked how can calcule the area of unusual shapes، and one of they scientists came up with a genius idea, which it say we must  devise this area of Some rectangles have the same width، and when we make this width become 0, and adding all this rectangle area,We will get An estimate of the area of this unusual shape
$(\sum_{} ^{} f(x) dx) $
So my question how they are   moved from $\sum_{} ^{} f(x) dx$ to $\int\limits_{}^{} f(x) dx$
I asked this question when I tried to solve this limit:$\lim_{n\to +\infty}(\frac{n! }{n^n})^{\frac{1} {n}} $ so  after that I have seen one idea about
use  the integral for solve it ...
So after this idea I think it  is wonderful because it may help in a lot of problems,
 A: The conceptual description of sums and integrals is the following.

*

*In a summation, $\sum_{n=a}^b f_n$, you are adding up the values of $f$ when evaluated at a finite set of points for $n$: $a$, $a + 1$, $a + 2$, ..., $b$.


*In an integration, $\int_{x=a}^b df(x)$, where $df(x) = f(x)\ dx$, you are adding up the values of $df$ at an uncountably infinite set, i.e. a set as big as the real numbers, of points: namely, every real number in the interval $[a, b]$. That is, if say, $[a, b] = [0, 1]$, you add up $df(0)$, $df(1)$, $df(0.5)$, also $df\left(\frac{1}{\pi}\right)$ and so many more that not only can we not write them all down, we can't even label them all using whole numbers only.
Note that it is very important that we integrated $f(x)\ dx$, not just $f(x)$, like in the sum. What $dx$ is, is a scaling factor, but not one that can be considered as being a real number. To understand its role, suppose that we consider a sum
$$\sum_{n=a}^b f_n \Delta n$$
. Here, $\Delta n$ is something that, on its own, outside the sum context does not have a defined value, but inside the sum, acquires or procures a value equal to the distance between successive steps of the $n$ variable, e.g. how much it increments in going from 2 to 3, 3 to 4, 4 to 5, and so on. But, as you can see, that's just $1$, i.e. $\Delta n = 1$. And multiplying by $1$ does nothing, so we can simply suppress it and write
$$\sum_{n=a}^b f_n$$
with no problems. The point is you can think it is "there", but hidden, so as to put the sum and integral notations on equal footing.
Now let us consider how we define the ideas of a "countable" and "uncountable" infinite sum of terms.
A countable infinite sum of terms is just a sum where $n$ is allowed to range from $a$ up to $\infty$, i.e. the usual infinite series. That is,
$$\sum_{n=a}^\infty f_n = f_a + f_{a+1} + f_{a+2} + f_{a+3} + \cdots$$
"forever". The trouble here is, of course, we have to define what we mean by "forever", and there's where we use a limit: we say that this sum is the limit as we add more and more terms on the end, i.e.
$$\sum_{n=a}^\infty f_n = \lim_{N\rightarrow\infty} \sum_{n=a}^N f_n$$
Now consider the idea of summing over all points in an interval, i.e. $x \in [a, b]$ and trying to write a
$$\sum_{x \in [a, b]} f(x)$$
. We cannot use the above definition because there are too many points: that's what "uncountable" means - if we plucked points out of the interval $[a, b]$ and labeled them "1st point", "2nd point", "3rd point", etc. we would actually run out of counting numbers before we ran out of real numbers in the interval! Hence, we cannot simply put them in some order and add up the terms as we did in the previous limit.
Instead, we need something more ingenious, and what that is is to take a limit over all possible finite samplings of points from the interval $[a, b]$, i.e. samples of some number $N$ of points $x_1, x_2, ..., x_N \in [a, b]$, which for ease we make such that $x_1 = a$, $x_N = b$, and ordered so that $x_1 < x_2 < \cdots x_{n-1} < x_n$, and we consider what happens as the samplings get denser and denser, i.e. both as $N$ goes up, and moreover as the largest interval between two adjacent such samples gets smaller. We write this as
$$\sum_{x \in [a, b]} f(x) := \lim_{||\Delta|| \rightarrow 0} \sum_{i=1}^n f(x_i)$$
("sum of an uncountably infinite number of terms" = "limit over finer and finer samplings from the interval".)
From this definition, you can then prove that the uncountable sum never converges unless $f$ is zero at all but a finite number of points!
Clearly, that's not particularly useful, and that's where $dx$ swoops in. We first ditch the sum notation for the integral notation,
$$\int_{x=a}^b f(x)$$
which means the same thing as the above, then we bring $dx$ into the picture to get
$$\int_{x=a}^b f(x)\ dx$$
And just as how that for finite sums $\Delta n$ took on the meaning of the increment, the same goes with the integral, except that in the "ultimate" limit $dx$ is too small to be a real number, and basically shrinks down all the uncountable things so they are not zero yet so small they can still after adding up amount to something finite.
But importantly, before the limit is reached, it is a real number, but one that's steadily shrinking as the sample gets denser. That is, it's just the increment between adjacent sample points - i.e. we define
$$\int_{x=a}^b f(x)\ dx := \lim_{||\Delta||\rightarrow 0} \sum_{n=1}^N f(x_i) \cdot \underbrace{(x_{i+1} - x_i)}_{\text{$\Delta x_i$ "approximation of $dx$"}}$$
Finally, you can visualize this as being that,at each stage, we have some rectangles of width $\Delta x_i$ and heights $f(x_i)$ and we are adding up their areas. And the limit as "adding up all the vertical lines at each point $x$ to get the whole region under the curve".
