Editing Integral (section)

=== Riemann integral ===
{{Main|Riemann integral}}
The Riemann integral is defined in terms of [[Riemann sum]]s of functions with respect to ''tagged partitions'' of an interval.<ref>{{Harvnb|Anton|Bivens|Davis|2016|pp=286−287}}.</ref> A tagged partition of a [[closed interval]] {{math|[''a'', ''b'']}} on the real line is a finite sequence

: <math> a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_{n-1} \le t_n \le x_n = b . \,\!</math>

This partitions the interval {{math|[''a'', ''b'']}} into {{mvar|n}} sub-intervals {{math|[''x''<sub>''i''−1</sub>, ''x''<sub>''i''</sub>]}} indexed by {{mvar|i}}, each of which is "tagged" with a specific point {{math|''t''<sub>''i''</sub> ∈ [''x''<sub>''i''−1</sub>, ''x''<sub>''i''</sub>]}}. A ''Riemann sum'' of a function {{mvar|f}} with respect to such a tagged partition is defined as

: <math>\sum_{i=1}^n f(t_i) \, \Delta_i ; </math>

thus each term of the sum is the area of a rectangle with height equal to the function value at the chosen point of the given sub-interval, and width the same as the width of sub-interval, {{math|Δ<sub>''i''</sub> {{=}} ''x''<sub>''i''</sub>−''x''<sub>''i''−1</sub>}}. The ''mesh'' of such a tagged partition is the width of the largest sub-interval formed by the partition, {{math|max<sub>''i''{{=}}1...''n''</sub> Δ<sub>''i''</sub>}}. The ''Riemann integral'' of a function {{mvar|f}} over the interval {{math|[''a'', ''b'']}} is equal to {{mvar|S}} if:<ref>{{Harvnb|Krantz|1991|p=173}}.</ref>

: For all <math>\varepsilon > 0</math> there exists <math>\delta > 0</math> such that, for any tagged partition <math>[a, b]</math> with mesh less than <math>\delta</math>,

: <math>\left| S - \sum_{i=1}^n f(t_i) \, \Delta_i \right| < \varepsilon.</math>

When the chosen tags are the maximum (respectively, minimum) value of the function in each interval, the Riemann sum becomes an upper (respectively, lower) [[Darboux integral|Darboux sum]], suggesting the close connection between the Riemann integral and the [[Darboux integral]].