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What does $dx$ mean?

So we're learning Riemann Integrals from 0 in my (Calculus? Pre-calculus?) class. Point is, we were taught the following notation:


Which seems to me pretty similar to the following expression


Which would at the same time resemble quite closely the sum that defines integrals (which, for us would be $\sum^n_{i=1}m_i(t_i-t_{i-1})$ or $\sum^n_{i=1}M_i(t_i-t_{i-1})$, where $m_i = \inf{}$ and $M_i = sup{}$ for their respective partitions).

Question is, does that $dt$ actually represent $(t_i-t_{i-1})$. If not, what does it represent, why is it there?

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marked as duplicate by Argon, joriki, Jasper Loy, Pedro Tamaroff, Henry T. Horton Oct 26 '12 at 2:48

This question was marked as an exact duplicate of an existing question.

That $dt$ is just a piece of notation that serves to remind us that an integral is a limit of Riemann sums, and that in a Riemann sum you have a term $(t_i - t_{i-1})$.

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So that means that I have to take the limit when the... sums tend to infinity in order to get the integral? – Julián Oct 26 '12 at 1:46

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