# Definite integral $\int ^b _a f(x)dx = \lim_{n \to \infty} \sum^n _{k=1} f(\zeta_k) \Delta x_k$

This is the mathematical description of the definite integral of $f(x)$ between $a$ and $b$:

$$\int ^b _a f(x)dx = \lim_{n \to \infty} \sum^n _{k=1} f(\zeta_k) \Delta x_k$$

In here, $\Delta x_k$ is the width of a rectangle, $f(\zeta_k)$ is the height of a rectangle, $\displaystyle \lim_{n \to \infty} \displaystyle \sum^n _{k=1}$ just is the sum of all of the n number of rectangles (which approaches $\infty$), $\displaystyle \int ^b _a f(x)$ just means the integral of the function f(x) from point a to b. What is left over in this expression which I cannot really give its use here, is the $dx$. What does it stand for here? I know it comes from the $\dfrac{dy}{dx}$ but I still don't really understand it here.

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You can also view $dx$ as a sort of limit of $\Delta x_k$ as $n$ grows. Then "every point $x$ is summed over". It abstract integration theory "$dx$" is a measure, and it can be more general than integration over, say, the real line. –  Simen K. Feb 25 '13 at 14:21
@SimenK. it suits as a answer more than comment. –  user45099 Feb 25 '13 at 14:26

The symbol "$dx$" serves as a label to indicate which variable is being integrated over.

It is perfectly fine to leave out "$dx$" if no confusion can arise. But consider the case where $f$ is a multivariate function: then the label is essential.

Moreover, we can interpret $dx$ as a kind of limit: The definition of the (Riemann) integral is a limit of the sum over the partition of $[a,b]$ as each $\Delta x_k$ (uniformly) goes to zero. This limit is independent of how the limit is taken, as long as each $\Delta x_k$ goes uniormly to zero. Thus, in the limit we have an "infinite sum" over "infinitesimally small intervals of lenth $dx$".

In abstract integraton theory, one may integrate over wildly different kinds of objects. Here, symbols like $dx$ stands for the abstract measure being used. It can, for example, include a weight function, such as Stieltjes integrals.

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I understand the first part of your answer, it just basically tells use which variable is integrated over, which can have its uses when you have multiple variables 'to choose from' (which isn't the case here ofc). I however don't understand what you mean by 'the definition sums over partitions of the interval [a,b] where x lives' –  Integrals Feb 25 '13 at 14:40
@Integrals I hope my edited version is better. –  Simen K. Feb 25 '13 at 19:40
The $d$ notation indicates the variable of integration. You can see similar roles for the variable $k$ of summation in $$\sum_{k = 1}^n a_k$$ and the variable $x$ of integration in $$\int_a^b f(x)\, dx.$$ Both variables are "dummy variables" that have scope confined to the inside of their operators (sum/integral). So, you can think of the $d$ as a prefix for the variable of integration.
The choice of $d$ does suggest it is a limit of the $\Delta x_k$ in a Riemann sum.