On the notation for integrals

Question

Possible Duplicate:
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:

$$\int^x_af(t)dt$$

Which seems to me pretty similar to the following expression

$$\sum^n_{i=1}f(t)(t_i-t_{i-1})$$

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?

Answer 1

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 $(t_i - t_{i-1})$.