As already noted, “$dx$ is just a placeholder in single-variable calculus. Ask a student what is being integrated in $\int f(x)\,dx$, and the answer will likely be “$f(x)$”. But this approach is at best misleading in multivariable calculus.

Consider the integral $\int\int(x^2+y^2)\,dy\,dx$. What is being integrated? If you said “$x^2+y^2$”, would you answer “$r^2$” for $\int\int r^3\,dr\,d\phi$, which is of course the same integral in polar coordinates? Reasonable arguments can be made that either $r^2$ or $r^3$ is being integrated, but such arguments miss the point; what is being added is in fact $r^3\,dr\,d\phi$.

The inclusion of the “measure” ($dx$, $dx\,dy$, $r\,dr\,d\phi$, …) as part of what is being added is a basic tool for the physicist, but it typically requires students to fundamentally change their notion of integration to be able to see it this way.