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

• deancodemo

#### deancodemo

This isn't really a homework question, but it has been bugging me for ages.
In $$\int f(x) \, dx$$ 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?

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.

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.

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)

Yes, yes. Another thing people tend to write is $$\lim_{\delta x \to 0} \frac{\delta y}{\delta x} = \frac{dy}{dx}$$. I guess this means that as $$\delta x$$ becomes very small, the quotient $$\frac{\delta y}{\delta x}$$ becomes exact. ie the derivative $$\frac{dy}{dx}$$.

dx is, in fact, a differential form. A one-form, to be exact.

- Warren

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.

One point- if you leave out the "dx" in, say, $\int f(x^2) dx$ so that you have only $\int sin(x^2)$, 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.

Also, $$\delta x$$ is small, but $$dx$$ is infinitely small.

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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

$$\int f(x) \, dx$$

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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There's a natural parallel between the summation and integral.
$$\int_a^b f(x) dx = \lim_{n \rightarrow \infty} \sum_{i = 0}^n f(x_i) \Delta x$$