The integral with oriented limits: $\int_a^a f := 0$ and $\int_b^a f := -\int_a^b f$

Why this item is first. The published definition of the integral does not cover this page. The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as $\sup_P L(f,P)$ and $\inf_P U(f,P)$, Darboux integrability as their equality, and the notation $\int_a^b f$ is stated for reals $a < b$, because the partitions it quantifies over are those of Partition of $[a,b]$ as a finite strictly increasing list $a = t_0 < t_1 < \dots < t_n = b$, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions, whose standing hypothesis is $a < b$: with $a = b$ the chain $a = t_0 < \dots < t_n = b$ is unsatisfiable. So $\int_a^b f$ is an undefined symbol whenever $a \ge b$, and every additivity statement below would be ill-formed as it is usually written. This item extends the notation, and nothing else: the object it names is still the Darboux integral of The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as $\sup_P L(f,P)$ and $\inf_P U(f,P)$, Darboux integrability as their equality, and the notation $\int_a^b f$.

Let $u, v \in \mathbb{R}$ and write

$[u \wedge v,\ u \vee v] \;:=\; \text{the closed interval with endpoints } u \text{ and } v$

(Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length). Let $f$ be a real-valued function whose domain contains that interval. Say that $f$ is integrable between $u$ and $v$ when either $u = v$, or $u \ne v$ and the restriction of $f$ to $[u \wedge v,\ u \vee v]$ is bounded (Lower bound, bounded below, bounded set) and Darboux integrable there (The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as $\sup_P L(f,P)$ and $\inf_P U(f,P)$, Darboux integrability as their equality, and the notation $\int_a^b f$, For bounded $f$ on $[a,b]$ and a partition $P$: the infimum $m_i$ and supremum $M_i$ of $f$ on the $i$-th subinterval, and the lower and upper Darboux sums $L(f,P) = \sum_i m_i \Delta_i$ and $U(f,P) = \sum_i M_i \Delta_i$). For such $f$ define

$\int_u^v f \;:=\; \begin{cases} \text{the Darboux integral of } f \text{ over } [u,v] & \text{if } u < v, \\[2pt] 0 & \text{if } u = v, \\[2pt] -\displaystyle\int_v^u f & \text{if } u > v. \end{cases}$

There is nothing to check for consistency. The three clauses are indexed by the three cases of trichotomy, $u < v$, $u = v$ and $u > v$, which are mutually exclusive and exhaustive; no pair of them ever applies to the same $(u,v)$. In particular the first clause is untouched, so on $u < v$ this is the published integral verbatim and every published theorem about it applies unchanged.

The middle clause is a stipulation, not a computation. It is not claimed that $0$ is a value forced by the $u < v$ definition in any limiting sense; that definition simply says nothing at $u = v$, and $\int_u^u f := 0$ is what is written there. It is also unconditional: no hypothesis on $f$ beyond being defined at $u$ is asked for, since the case $u = v$ never refers to a partition.

The two consequences used throughout the page

Antisymmetry, for every pair. For all reals $u, v$ with $f$ integrable between them,

$\int_u^v f \;=\; -\int_v^u f .$

Indeed if $u < v$ then $v > u$ and the third clause reads $\int_v^u f = -\int_u^v f$, which rearranges to the display; if $u = v$ both sides are $0$; and if $u > v$ the third clause is the display itself.

Absolute values agree. Consequently $\bigl|\int_u^v f\bigr| = \bigl|\int_v^u f\bigr|$ for every such pair.

An obligation recorded here and discharged elsewhere. With this convention the additivity identity

$\int_u^v f \;+\; \int_v^w f \;=\; \int_u^w f$

holds for every arrangement of $u, v, w$ in an interval on which $f$ is integrable, not only for $u < v < w$. That is a theorem and not part of this definition; it is proved as the last clause of For $a<c<b$: $f$ is integrable on $[a,b]$ if and only if it is integrable on $[a,c]$ and on $[c,b]$, and then $\int_a^b f = \int_a^c f + \int_c^b f$; with the oriented form for arbitrary $a,b,c$, and nothing on this page uses it before it is proved there.

• This is notation, and it is a real notation. Without it the substitution theorem could not be stated with the limits $\varphi(c)$ and $\varphi(d)$ in the order the map produces them, since a differentiable $\varphi$ need be neither injective nor monotone; and the integral function $x \mapsto \int_a^x f$ would be undefined at $x = a$.

• One published inequality is not orientation-invariant, and that is a trap. The estimate $\bigl|\int_u^v f\bigr| \le \int_u^v |f|$ is guaranteed only for $u \le v$: at $u > v$ the right-hand side is $-\int_v^u |f| \le 0$ while the left-hand side is $\ge 0$, so the inequality fails whenever $\int_v^u |f| > 0$. The form valid for every pair is $\bigl|\int_u^v f\bigr| \le \bigl|\int_u^v |f|\bigr|$, and this is stated where it is proved (If $f,g$ are integrable on $[a,b]$ then so are $\lvert f\rvert$, $f^{2}$, $fg$, $\max(f,g)$ and $\min(f,g)$, and $\bigl\lvert\int_a^b f\bigr\rvert \le \int_a^b\lvert f\rvert$).

• Integrability is a property of the unordered pair. By construction, $f$ is integrable between $u$ and $v$ if and only if it is integrable between $v$ and $u$, since both refer to the same closed interval; only the sign of the value remembers the order.