Editing Riemann integral (section)

=== Riemann sum ===
{{Main|Riemann sum}}
Let {{mvar|f}} be a real-valued function defined on the interval {{math|[''a'', ''b'']}}. The '''Riemann sum''' of {{mvar|f}} with respect to a tagged partition {{math|''P''(''x'', ''t'')}} of {{math|[''a'', ''b'']}} is<ref>{{Cite book |last=Krantz |first=Steven G. |url=https://www.worldcat.org/oclc/56214595 |title=Real Analysis and Foundations |date=2005 |publisher=Chapman & Hall/CRC |isbn=1-58488-483-5 |location=Boca Raton, Fla. |oclc=56214595 |page=173}}</ref>
<math display="block">\sum_{i=0}^{n-1} f(t_i) \left(x_{i+1}-x_i\right).</math>

Each term in the sum is the product of the value of the function at a given point and the length of an interval. Consequently, each term represents the (signed) area of a rectangle with height {{math|''f''(''t<sub>i</sub>'')}} and width {{math|''x''<sub>''i'' + 1</sub> − ''x<sub>i</sub>''}}. The Riemann sum is the (signed) area of all the rectangles.

Closely related concepts are the ''lower and upper Darboux sums''. These are similar to Riemann sums, but the tags are replaced by the [[infimum and supremum]] (respectively) of {{mvar|f}} on each sub-interval:
<math display="block">\begin{align}
L(f, P) &= \sum_{i=0}^{n-1} \inf_{t \in [x_i, x_{i+1}]} f(t)(x_{i+1} - x_i), \\
U(f, P) &= \sum_{i=0}^{n-1} \sup_{t \in [x_i, x_{i+1}]} f(t)(x_{i+1} - x_i).
\end{align}</math>

If {{mvar|f}} is continuous, then the lower and upper Darboux sums for an untagged partition are equal to the Riemann sum for that partition, where the tags are chosen to be the minimum or maximum (respectively) of {{mvar|f}} on each subinterval. (When {{mvar|f}} is discontinuous on a subinterval, there may not be a tag that achieves the infimum or supremum on that subinterval.) The [[Darboux integral]], which is similar to the Riemann integral but based on Darboux sums, is equivalent to the Riemann integral.