We say that $I \in \mathbb{R}$ is
the (Euler) integral of $f$ over $D=[a,b]$, denoted $\int_D f$, $\int_a^b f$, or $\int_a^b f(x)dx$
if for any choices of $C_N$, the limit of $S_N$ as $N \to \infty$ is $I$, $$ \lim_{N \to \infty} S_N = I.$$
I.e., for any $\epsilon >0$ , there is an $M$, so that for any $N>M$ and any choices of $C_N$, $|S_N -I| < \epsilon$ .
We can break the analytic definition for the integral into three steps needed to verify it being correctly applied.
- "for any $ \epsilon \gt 0$": Consider a positive real
number, $\epsilon$, that is arbitrarily chosen and usually considered
to be small, that is "close to $0$".
The number $\epsilon$ will be a control on the size of $N$, and thereby
control $\Delta x$ ensuring that values of $S_N$ are "close to $I$".
- "there exists a number $M $": Now
based on the chosen $\epsilon$, the task is to find the number, $N$.
The number $N$ provides a control on the
number $\Delta x$ and this the partition of $P = \{x_i\}$ of the domain
of $f$ to keep $c_i$ limited so that $\Sigma_1^Nf(c_i)\Delta x=S_N$ is close to $I$".
In fact the constraint $N > M$ amounts to having $S_N$ in the interval $(I-\epsilon,I+\epsilon)$. - "if $N>M$ then $|S_N - I| < \epsilon$": The number $N$ works in the definition for the given $\epsilon$ only if when $N>M$ then it can be demonstrated that
$|S_N-I| < \epsilon$. The demonstration will connect the
constraint on $N$ with the actual function's definition and an analysis
that eventually results in a conclusion connecting to the inequality $|S_N-I| < \epsilon$.
Definition.2.(Arbitrary Partition) : Suppose $P=\{a=x_0\le x_1\le x_2 ... \le x_N=b\}$ and $x_k-x_{k-1}= \Delta x_k$ for $ k = 1,2, ... N$,
and choose $C_P=\{c_k \in [x_{k-1},x_k], k = 1,...,N\}.$ Denote by
$||P||=||\Delta x|| = max\{\Delta x_k\}$, the mesh of the
partition $P$.
Let $S_P= \Sigma_{k=1}^{k=N} f(c_k) \Delta x_k$ , a sum that depends on the choice of $C_P$ for each $P$.
We say that $I \in \mathbb{R}$ is the (Riemann) integral of $f$
over $D=[a,b]$, denoted $\int_D f$, $\int_a^b f$, or $\int_a^b f(x)dx$
if for any choices of $P$ and $C_P$, the limit of $S_P$ as $||P|| \to 0$
is $I$, $$ \lim_{||P|| \to 0} S_P = I.$$ I.e., for any $\epsilon >0$
, there is an $\delta>0$, so that for any $P$ with $||P||<
\epsilon$ and any choices of $C_P$, $|S_P -I| < \epsilon$