Which one of these is correct, are both of these right? $$\Delta x_i = (x_i-x_{i-1}) $$
$$ \int_a^b f(x) \, dx = \lim_{\|x\| \rightarrow 0} \ \sum_{i=1}^n f(x_i)\Delta x_i $$
$$ \int_{x_0}^{x_n} f(x) \, dx = \lim_{n \rightarrow ∞} \sum_{i=1}^n f(x_i)\Delta x_i $$
Surely this one has an infinite upper bound and does not express the number of partitions going towards infinity. Which notation is correct to express the Riemann Sum in terms of an integral, i think i have confused my self. I have hardly done any calculus just so you know.