Editing Integral (section)

=== Conventions ===
In this section, {{mvar|f}} is a [[Real number|real-valued]] Riemann-integrable [[Function (mathematics)|function]]. The integral

: <math> \int_a^b f(x) \, dx </math>

over an interval {{math|[''a'', ''b'']}} is defined if {{math|''a'' &lt; ''b''}}. This means that the upper and lower sums of the function {{mvar|f}} are evaluated on a partition {{math|''a'' {{=}} ''x''<sub>0</sub> ≤ ''x''<sub>1</sub> ≤ . . . ≤ ''x''<sub>''n''</sub> {{=}} ''b''}} whose values {{math|''x''<sub>''i''</sub>}} are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating {{mvar|f}} within intervals {{math|[''x''<sub> ''i''</sub> , ''x''<sub> ''i'' +1</sub>]}} where an interval with a higher index lies to the right of one with a lower index. The values {{mvar|a}} and {{mvar|b}}, the end-points of the [[Interval (mathematics)|interval]], are called the [[limits of integration]] of {{mvar|f}}. Integrals can also be defined if {{math|''a'' &gt; ''b''}}:''<ref name=":1" />''

:<math>\int_a^b f(x) \, dx = - \int_b^a f(x) \, dx. </math>

With {{math|''a'' {{=}} ''b''}}, this implies:

:<math>\int_a^a f(x) \, dx = 0. </math>

The first convention is necessary in consideration of taking integrals over subintervals of {{math|[''a'', ''b'']}}; the second says that an integral taken over a degenerate interval, or a [[Point (geometry)|point]], should be [[0 (number)|zero]]. One reason for the first convention is that the integrability of {{mvar|f}} on an interval {{math|[''a'', ''b'']}} implies that {{mvar|f}} is integrable on any subinterval {{math|[''c'', ''d'']}}, but in particular integrals have the property that if {{mvar|c}} is any [[Element (mathematics)|element]] of {{math|[''a'', ''b'']}}, then:''<ref name=":0" />''

:<math> \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx.</math>

With the first convention, the resulting relation

: <math>\begin{align}
 \int_a^c f(x) \, dx &{}= \int_a^b f(x) \, dx - \int_c^b f(x) \, dx \\
 &{} = \int_a^b f(x) \, dx + \int_b^c f(x) \, dx
\end{align}</math>

is then well-defined for any cyclic permutation of {{mvar|a}}, {{mvar|b}}, and {{mvar|c}}.