TITLE: What exactly is a 'dummy variable'?
QUESTION [8 upvotes]: I was watching an online course (The Calculus You Need - MIT OpenCourseWare), when (around 03:08), the lecturer (Gilbert Strang) says that he doesn't "care what that dummy variable is" (the variable x associated with the function y). He made the following change in the video:
$$\frac{d}{dx}\int_{0}^{x}y(x) = y(x)\Rightarrow \text{ change } \Rightarrow\frac{d}{dx}\int_{0}^{x}y(t)dt = y(x)$$
I don't know why the notation of that variable doesn't cause any ambiguity here (I'm assuming Strang wanted to mean that), once x is clearly creating a dependency relation with the bound and the variable associated with the function y.
Ultimately... why it doesn't make a difference to call the variable associated with y by x (in this case)? What exactly is a 'dummy variable'?

REPLY [3 votes]: To add to what was already said, I find it helpful to think in programming terms. What is a dummy variable? It's a local variable with very limited scope - a sum, integral, or other similar construct - and it had better not appear anywhere outside that. If it does, as in that initial expression $\frac{d}{dx}\int_0^x y(x)$, we'll get errors and unpredictable results. For example, here? In order to get something that parses, we have to take that integral with respect to some other variable - which means that inside the integral, $x$ and $y(x)$ are constants. We then get $\int_0^x y(x)\,dt=xy(x)$, so its derivative with respect to $x$ is $y(x)+xy'(x)$, and the solution of the equation is for $y(x)$ to be a constant function.
The equation after the change is clearly what was intended all along, and it doesn't mean the same thing as the equation before the change. It doesn't matter what we call the dummy variable - as long as we're careful not to conflict it with a variable name on the outside and cause out-of-scope errors.
As for leaving the $d?$ off the integral? I'm not a fan of that; it's a delimiter, which clearly marks what we're integrating, and what variable we're integrating with respect to. As far as I'm concerned, it's part of the integral sign. The only way I'd ever leave it off is if the dummy variable is left out entirely. $\int_a^b f(x)\,dx$ or $\int_a^b f$ are OK and unambiguous; $\int_a^b f(x)$ is malformed.