# Riemann Integral **An [[integral]] equal to the limit of a sequence of sums of the areas of vertical rectangular approximations of the integral as the width of the rectangles approach zero.** ![[RiemannIntegral.svg]] *A sequence of Riemann sums; the limit of the sequence as the width of the partitions approaches zero is the Riemann integral* The *Riemann integral* is a definition of the integral of a function. It is defined as the convergent value of a sequence of sums of the areas of vertical rectangles formed by a *partition of the integral* as the widths of the sub-intervals approaches zero. These sums are known as *Riemann sums*. Riemann integrals are typically evaluated by *numerical integration* or the [[fundamental theorem of calculus]]. ## Definition ### Partitions > [!Partition of an interval] > A *partition* of an interval $[a, b]$ is a sequence of numbers such that if $a=x_{0}$ and $b = x_{n}$, > $x_{0}<x_{1}<x_{2}<\cdots<x_{i}<\dots <x_{n}$ > A pair of adjacent numbers form a *sub-interval* $[x_{i},x_{i+1}]$, the length of which is $\Delta_{i}=x_{i+1}-x_{i}$. > > The *mesh* of a partition is the *length of the longest sub-interval*, and is denoted $\delta$. A *tagged partition* is a partition in which each sub-interval is assigned a *sample point*, that is, for each sub-interval $[x_{i}, x_{i+1}]$, it is tagged with a number $x_{i} \le t_{i} \le x_{i+1}$. The partition used *does not* need to create *equal length* sub-intervals and the Riemann integral gives the correct value as long as the width of each sub-interval approaches zero. ### Riemann sum > [!Riemann sum] > Let $f$ be a real-valued function defined on a *closed interval*. > > The *Riemann sum* of $f$ is, with respect to a tagged partition of $f$, > $\sum_{i=0}^{n-1}f(t_{i})\Delta_{i}$ ![[RiemannSum.svg|500]] *A Riemann sum with the representation of a single term highlighted* A Riemann sum consists of a term for *each sub-interval* $[x_{i}, x_{i+1}]$ equal to the product of the *value* of the function *at the sample point* $f(t_{i})$ of the sub-interval and the *width* of the interval $\Delta_{i}$. Thus, each term represents the signed area of a rectangle and the entire Riemann sum is an *approximation of the signed area* between the graph of $f$ and the $x$-axis. ### Riemann integral > [!Riemann integral] > Let $f$ be a real-valued function defined on a *closed interval*. > > The *Riemann integral* of $f$ is the limit of the Riemann sums of $f$ as the mesh of the tagged partitions approaches zero. > $\int_{a}^{b}f(x)\;dx=\lim_{\delta\rightarrow0}\sum_{i=0}^{n-1}f(t_{i})\Delta_{i}$ ## Integrability For a Riemann integral to exist, the mesh of tagged partitions of the function must be able to tend towards zero. > [!Riemann integrability] > A function is *Riemann-integrable* if, for all $\varepsilon > 0$, there exists some $\delta > 0$ such that for any tagged partition of the function whose mesh is less than $\delta$, > $\left|\left(\sum_{i=0}^{n-1}f(t_{i})\Delta_{i}\right)-s\right|<\varepsilon$ > If the function is Riemann-integrable, then its Riemann integral is equal to $s$. An alternate characterisation of functions as Riemann-integrable or not is the *Lebesgue-Vitali theorem*. > [!NOTE] Lebesgue-Vitali theorem > The *Lebesgue-Vitali theorem* states that a *bounded function* on a *closed interval* is Riemann-integrable if and only if it has *countably many [[Discontinuity|discontinuities]]*.