This is also connected to the visualization of the sums used in the definition. Mapping diagrams can play an important supporting role in visualizing the definite integral, also connected to the defining sums.

More generally, when

and choose $C_N=\{c_k \in [x_{k-1},x_k], k = 1,...,N\}.$

Let $S_N= \Sigma_{k=1}^{k=N} f(c_k) \Delta x$ , a sum that depends on the choice of $C_N$ for each $N$.

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.

**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 $C_P$, $|S_P -I| < \epsilon$

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 $|S_N-I| < \epsilon$.

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 $C_P$, $|S_P -I| < \epsilon$

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 $||P||$ ensuring that values of $S_P$ are "close to $I$".

**"****there is an $\delta>0$,":**Now based on the chosen $\epsilon$, the task is to find the number, $\delta >0$. The number $\delta$ provides a control on the number $||P||$ and thus on the partition of $P$ of the domain of $f$ to keep $c_k$ limited so that $\Sigma_1^Nf(c_k)\Delta x_k=S_P$ is close to $I$".

In fact the constraint $||P|| < \delta $ amounts to having $S_P$ in the interval $(I-\epsilon,I+\epsilon)$.**"if****$||P||<\delta$ then $|S_P - I| < \epsilon$****":**The number $\delta$ works in the definition for the given $\epsilon$**only if**when $||P||<\delta$ then it can be demonstrated that $|S_P-I| < \epsilon$. The demonstration will connect the constraint on $||P||$ with the actual function's definition and an analysis that eventually results in a conclusion connecting to the inequality $|S_P-I| < \epsilon$.