I have encountered the following integral:
$\int_{x-d}^{x+d}f(y)dy$
I am trying to figure out what is the role of $d$ in this integral. Is the $d$ at the beginning of the integral the same as the $dy$? If so, why do we need to integrate from $x-d$ to $x+d$? Or is it just some other variable which has been poorly named?
Some people have the habit of using non-italicized $\rm d$ before integration constant: $$ \int_{x-d}^{x+d}f(y)\,\mathrm{d}y $$ I never understood why, but perhaps on this occasion this would be useful. The font emphasizes that $\mathrm{d}$ is not a variable, but a kind of operator applied to a variable (namely $y$), which is loosely described as "infinitesimal change".
Of course, one can also say that $\mathrm{d}y$ is a two-letter mathematical symbol in which the letters have no individual meaning.
And if you think this is confusing notation, wait till you come across a multiple integral over the space of matrices $$\begin{pmatrix} a & b \\c & d\end{pmatrix}$$ like $\int\int\int\int \dots da\,db\,dc\,dd$.