Intuition About dx in Integral Notation

Question

So, an integral is notated like this:

$$\int_a^bf(x)dx$$

And from my understanding, it's an operator that is defined for three operands: $a$ and $b$, which can be anything, and an integrand of the form $f(x)dx$.

$dx$ is just an infinitesimal number, so $f(x)dx$ is simply $f(x)$ multiplied by $dx$. This gives an infinitesimal based on $x$, and the integral is the infinite sum of all of those. I found a good page that explains this idea nicely.

But what about integrands that are not of the form $f(x)dx$? For example, things like:

• $\int_a^b2$
• $\int_a^bf(x)dy$
• $\int_a^b(f(x)dxdy)$
• $\int_a^bf(x)$

Certainly, if the intuition explained above is true (i.e: $f(x)dx$ is simply an algebraic expression; therefore the integral operator must accept any algebraic expression as its integrand), then these things must be syntactically legal. Is that wrong? How should these things be interpreted?

I suspect that my interpretation of $f(x)dx$ is wrong. $dx$ is part of the integral's notation, and plays a special role in what the integral operator does (defining the variable of integration). But $dx$ is also supposed to represent a numerical value which is multiplied by $f(x)$. How can it be both of those things at once?

I'm probably overthinking this. I just want to understand the notation and the intuition behind it. I hope someone can recognize what my confusion is and rectify it for me.

Answer 1

The integral symbol is simply a "nick name" for writing a more complex limit. $ \int_a^b f(x) dx = \lim_{n \to \infty} \sum^n_{i=1}f(c_i)\Delta x_i $. The "dx" bit tells us which variable we are integrating against it isn't really a number or a variable in its own right.

Answer 2

The pure mathematicians in the group seem to be leaning heavily toward the idea that this is just an artifact of notation and that there's no real factor being multiplied there. There is something to this view, so I don't want to come off as going against it completely. I'm not echoing points from other answers, but see, for example, the answer by Q the Platypus that talks about the Reimann sum.

From a physics point of view, however, I think we have to assign at least a modicum of meaning to it from the point of view of dimensional analysis. Take a simple equation like

$x(t) = x_0 + \int_0^t v(t') dt'$

for velocity $v(t)$ and position $x(t)$ with initial position $x_0$. The units on that will only work out if you assign units of time to the $dt$ "factor" that are consistent with the units used to measure $v$. At minimum it's also representing a requirement for consistency in measurement between the integrand, the limits of integration, and the units of measure in the result. In some sense it is still just notation, but - insofar as you asked about intuition - this definitely showing you a bit more than just which variable is integrated.