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I can't for the life of me seem to decide if using the variable of integration as a bound makes sense. For instance, integrating $y=x$ from $0$ to $x$. I don't think it does… But I'm not sure.

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    Letters/variables in mathematical expressions are either “bound” or “free.” Free variables are those you can “plug in a value” for. Bound variables are those you can’t. You can rename a bound variable without changing the meaning of the mathematics. $\int_0^x{x,dx}$ probably means $\int_0^\color{purple}{x}{\color{red}{x},d\color{red}{x}}$. There is one $x$, the red one, that’s bound, and an unrelated one, the purple one, that’s free. It’s confusing, but possible to make sense out of, because the upper limit of a single integral can’t be bound to the “variable of integration.” – Steve Kass Apr 18 '14 at 00:26
  • https://math.stackexchange.com/questions/3533541/integral-notation-confusion-x-vs-t?noredirect=1 – Mike Earnest Sep 13 '23 at 17:48

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It is best practice to use a dummy variable in the integrand itself. For example, we could write $$\int_0^x{t\,dt}=\frac{x^2}{2}$$ It does not really make sense to write something like $$\int_0^x{x\,dx}$$ because $x$ is a variable which is changing as the integral is computed. $x$ starts at $0$ and moves toward... toward $x$? That said, I think most people would agree that the value of that integral should be considered as $x^2/2$. Just note that the $x$ in the integral bound and the $x$'s in the integrand are technically different.

Perhaps it will be illumating to consider the discrete analogue. What would $$\sum_{n=0}^n{n}$$ mean?