Integration

Reimann Sums

We divide a interval $[a,b]$ into $n$ sub-intervals, not necessarily of same size, such that we have values of some $x$ in between $a,b$ that define as $$ x_{0} = a < x_{1}< x_{2}<x_{3}\dots < x_{n-1} < x_{n} = b $$and the intervals are defined as from $[x_0,x_1],[x_1,x_2]$ and so on.
All the points where we split the original interval, ie $x_0,x_1,x_2\dots$ are called, together, the Partition of $[a,b]$
In each interval if we consider a point $c_k$, and draw a rectangle with height $f(c_k)$ and width equal to the width of the partition, ie, $w=x_k-x_{k-1}$, and we sum all the areas of the rectangle up, (preserving sign of $f(c_k)$), we get the Reimann Sum, $$ S = \sum_{k=1}^n f(c_{k})\cdot\Delta x_{k} $$

For all the intervals, the maximum width of intervals is known as Norm of the partition and is represented as $||P||$

Definite integral

Let $f(x)$ be a function on closed interval $[a,b]$, We say that the number $J$ is the definite integral of $f$ over $[a,b]$, and $J$ is the limit of Reimann sums if the following condition is met

Given any number $ϵ>0$, there is a corresponding number $\delta >0$, such that for every partition $P$, of $[a,b]$, with $||P||<\delta$, and any choice of $c_k$ in $[x_{k-1},x_k]$, we have $$ \left|\sum_{k-1}^n f(c_{k})\cdot \Delta x_{k}\right| < \epsilon $$

When this condition is satisfied, we say that the Reimann sums of $f$ converge on $[a,b]$ to $J$, and that $f$ is Integrable on $[a,b]$