In some elementary introductions to integration I have seen the Riemann integral defined in terms of the limit of the following sum $$\int_{a}^{b}f(x)dx:=\lim_{n\rightarrow\infty}\sum_{i=1}^{n}f(x^{\ast}_{i})\Delta x$$ where the interval ##[a,b]## has been partitioned such that ##a=x_{1}<x_{2}<\cdots <x_{n-1}<x_{n}=b##, with ##x^{\ast}_{i}\in [x_{i-1},x_{i}]## (and arbitrary point in the sub-interval ##[x_{i-1},x_{i}]##), and ##\Delta x=\frac{b-a}{n}## the width of each sub-interval.(adsbygoogle = window.adsbygoogle || []).push({});

However, I have also seen it defined in terms of a similar sum, differing in the fact that the width of each sub-interval does not have to be equal. Indeed, given the same partition, we have that ##\text{Max}\,\lvert\Delta x_{i}\rvert=\lbrace \lvert x_{2}-x_{1}\rvert,\ldots , \lvert x_{i}-x_{i-1}\rvert,\ldots , \lvert x_{n}-x_{n-1}\rvert\rbrace##, and then $$\int_{a}^{b}f(x)dx:=\lim_{n\rightarrow\infty}\lim_{\text{Max}\,\lvert\Delta x_{i}\rvert\rightarrow 0}\sum_{i=1}^{n}f(\zeta_{i})\Delta x_{i}$$ where ##\Delta x_{i}=x_{i}-x_{i-1}## is the width of the ##i^{th}## sub-interval and ##\zeta_{i}\in [x_{i},x_{i-1}]## is an arbitrary point within this interval.

I have a couple of questions about these definitions:

1. Is the second definition I gave in some sense "better" than the first (in the sense that it is more general - it allows for each of the sub-intervals to have a different length)?

2. It is my understanding that ##\text{Max}\,\lvert\Delta x_{i}\rvert## denotes the maximum width of a given sub-interval. Given this, why do we take the limit as this maximum width tends to zero (especially as the width of each interval in the sum isn't necessarily the maximum)? Is the intuition that we wish for the width of each interval to become infinitesimally small as the number of sub-intervals becomes infinitesimally large (hence taking the limit ##n\rightarrow\infty##), and thus, by taking the limit as ##\text{Max}\,\lvert\Delta x_{i}\rvert\rightarrow 0## this ensures that the width of each interval will become infinitesimally small (regardless of whether ##x_{i}-x_{i-1}=\text{Max}\,\lvert\Delta x_{i}\rvert## or whether it is smaller than the maximum width)?!

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# I Definitions of the Riemann integral

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