What *exactly* does the 'dx' mean in integral notation?

In summary, the 'dx' in the integral notation, \int f(x) \, dx, represents an infinitesimally small increment of the variable x and is used to specify which variable is being integrated with respect to. In physics, it is commonly used to find the area under a curve, while in later courses like real analysis, it can be left out. In differential geometry, it takes on a more powerful form as a differential form. This notation also ties into the concept of infinitesimals in calculus, where dx represents an infinitely small change in x. Overall, the 'dx' in integral notation serves as a way to denote the unit of measure in an integral, similar to how it is used in summation
  • #1
deancodemo
20
0
This isn't really a homework question, but it has been bugging me for ages.
In [tex]\int f(x) \, dx[/tex] what exactly does the 'dx' represent? Is it a differential? What is a differential?

I only use the 'dx' part to identify the variable that the function is being integrated with respect to... or does it have another meaning?
 
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  • #2
In the kind of math we do in physics, it often represents an infinitesimally small increment of the variable x. The classic example is finding the area under a curve: you divide the area up into a bunch of rectangles such that the rectangle centered at x has height f(x) and width dx. Then the area of the rectangle is just the thing that appears inside the integral, f(x)dx, and the operation of integration itself becomes little more than a sum. (I believe this is called a Riemann integral) Of course, the equations and applications can get a lot more complicated than that, but still, in physics it's usually possible to think of dx as that kind of infinitesimal increment of x.
 
  • #3
According to my former real analysis teacher, it is nonsense. And don't get him started on L'Hospital's rule!

In calculus you would use it like you said, to specify which variable you are integrating with respect to. In later courses like real analysis, theory of metric spaces, etc, you usually leave it out.
 
  • #4
It means that you are taking the integral of something with respect to the change in x, where the change in x = lim x->infinity (delta x)
 
  • #5
Yes, yes. Another thing people tend to write is [tex]\lim_{\delta x \to 0} \frac{\delta y}{\delta x} = \frac{dy}{dx}[/tex]. I guess this means that as [tex]\delta x[/tex] becomes very small, the quotient [tex]\frac{\delta y}{\delta x}[/tex] becomes exact. ie the derivative [tex]\frac{dy}{dx}[/tex].
 
  • #6
dx is, in fact, a differential form. A one-form, to be exact.

Read the "Special cases" section.

- Warren
 
  • #7
Originally (not counting non-standard analysis) it was used just to denote the variable(s) that one was integrating over. As such, it is left out in many analysis texts as unnecessary, but it comes back in a much more powerful form as the notion of a differential form, when you go on to study differential geometry.
 
  • #8
One point- if you leave out the "dx" in, say, [itex]\int f(x^2) dx[/itex] so that you have only [itex]\int sin(x^2)[/itex], you might wind up using the substitution u= x^2 and not recognize that it is a mistake.

Essentially, an integral is a 'measurement' and dx tells the size of the unit of measure.
 
  • #9
I found a good wiki page about this topic http://en.wikipedia.org/wiki/Differential_(infinitesimal)" [Broken].

Also, [tex]\delta x[/tex] is small, but [tex]dx[/tex] is infinitely small.
 
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  • #10
From http://en.wikipedia.org/wiki/Differential_(infinitesimal)" [Broken]:

Differentials are also used in the notation for integrals because an integral can be regarded as an infinite sum of infinitesimally small quantities: the area under a graph is obtained by subdividing the graph into infinitesimally thin strips and summing their areas. In an expression such as

[tex]\int f(x) \, dx[/tex]

the integral sign (which is a modified long s) denotes the infinite sum, whereas the differential dx denotes the infinitesimally thin strips.
 
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  • #11
There's a natural parallel between the summation and integral.
[tex]\int_a^b f(x) dx = \lim_{n \rightarrow \infty} \sum_{i = 0}^n f(x_i) \Delta x[/tex]
 

1. What is the purpose of the 'dx' in integral notation?

The 'dx' in integral notation represents the variable of integration, which indicates the variable with respect to which the integral is being evaluated. It is an essential part of the notation and helps to specify the limits of integration.

2. How is the 'dx' related to the concept of calculus?

The 'dx' is a crucial element in the concept of calculus, as it represents an infinitesimal change in the independent variable of a function. In integration, it helps to calculate the area under a curve by dividing it into infinitely small rectangles.

3. Can the 'dx' be replaced with any other letter?

Yes, the 'dx' can be replaced with any other letter, such as 'dt' or 'dy', depending on the variable of integration. This allows for the integration of functions with multiple variables.

4. Is the 'dx' always written at the end of the integral?

In most cases, the 'dx' is written at the end of the integral, but it can also be written at the beginning or in between the integrand and the limits of integration. However, the placement of the 'dx' does not affect the value of the integral.

5. What happens if the 'dx' is omitted in an integral?

If the 'dx' is omitted, the integral is considered incomplete and cannot be evaluated. The 'dx' is a crucial part of the notation and must be included to specify the variable of integration and calculate the value of the integral.

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