The Riemann Integral

The capstone of the one-variable foundations (analysis spine). We define the integral via Darboux upper/lower sums, characterize which functions are integrable, and prove the Fundamental Theorem of Calculus linking integration and differentiation. The change-of-variables and integration-by-parts rules proved here are the one-dimensional ancestors of the multiple-integral and Stokes theorems later in the spine.

References: Rudin, Principles of Mathematical Analysis, ch. 6; Spivak, Calculus.

1. Partitions and Darboux Sums

Let $f:[a,b]\to \mathbb{R}$ be bounded. A partition is a finite set $P=\{a=x_0<x_1<\cdots <x_n=b\}$. On each subinterval $[x_{i-1},x_i]$ put

$$m_i=\underset{[x_{i-1},x_i]}{\inf }f,M_i=\underset{[x_{i-1},x_i]}{\sup }f$$

(these exist by boundedness and completeness, real-numbers.md). The lower and upper Darboux sums are

$$L(f,P)=\sum _{i=1}^n{m}_i\Delta x_i,U(f,P)=\sum _{i=1}^n{M}_i\Delta x_i,\Delta x_i=x_i-x_{i-1}.$$

Refining a partition raises $L$ and lowers $U$, and every lower sum is $\le$ every upper sum. Hence the lower and upper integrals

$$\underline{{\int }_a^b}f=\underset{P}{\sup }L(f,P)\le \overline{{\int }_a^b}f=\underset{P}{\inf }U(f,P)$$

are well-defined.

Existence vs. matching — two different questions. The lower and upper integrals always exist as real numbers: for bounded $f$ the set of lower sums is bounded above (by any upper sum), so its supremum exists by completeness (real-numbers.md §2), and dually for the infimum. What is not guaranteed is that they are equal — that is an extra property of $f$, not a theorem about all functions. The sup/inf already hide a limiting process (no separate $\lim$ is needed; the explicit mesh-limit formulation in §2 is equivalent), but completeness only delivers the two numbers, not their coincidence. The standard failure is the Dirichlet function $f=1_{\mathbb{Q}}$ on $[0,1]$: every subinterval meets both $\mathbb{Q}$ and its complement, so $m_i=0$ and $M_i=1$ for every partition, giving $\underline{{\int }_0^1}f=0\ne 1=\overline{{\int }_0^1}f$. It is bounded but not integrable. Matching is exactly what §3 must prove for nice functions.

2. Riemann Integrability

$f$ is Riemann (Darboux) integrable on $[a,b]$ if the upper and lower integrals coincide; the common value is

$${\int }_a^{b}f(x)dx.$$

Riemann criterion. $f$ is integrable iff for every $\epsilon >0$ there is a partition $P$ with $U(f,P)-L(f,P)<\epsilon$.

This is the practical test for integrability and the analogue of the Cauchy criterion (sequences-series.md): it never mentions the value of the integral.

2.1 The Riemann construction (tagged sums)

So far the integral has been built the Darboux way (§1) — via $\sup$/$\inf$ of lower/upper sums. Riemann's original construction, which gives the page its name, instead samples $f$ at chosen points. A tagged partition is a partition $P=\{x_0<\cdots <x_n\}$ together with tags $t_i\in [x_{i-1},x_i]$, and its Riemann sum is

$$S(f,P,\{t_i\})=\sum _{i=1}^{n}f(t_i)\Delta x_i.$$

Write the mesh $\Vert P\Vert ={\max }_i\Delta x_i$. Then $f$ is Riemann integrable if there exists a number $I$ such that

$$\forall \epsilon >0\exists \delta >0:\Vert P\Vert <\delta ⟹|S(f,P,\{t_i\})-I|<\epsilon$$

for every choice of tags. Such an $I$, if it exists, is unique (two candidates $I,I^{\prime }$ would satisfy $|I-I^{\prime }|<2\epsilon$ for all $\epsilon$, forcing $I=I^{\prime }$), so it is legitimate to define the integral to be that value,

$${\int }_a^{b}f(x)dx:=I.$$

Intuitively, once the partition is fine enough, the sum of "height $\times$ width" rectangles is within $\epsilon$ of $I$ no matter where each height is sampled — the literal "area under the graph as a limit of rectangle sums." This is the same number produced by the Darboux definition of §2, by the equivalence below.

Equivalence (Riemann = Darboux). $f$ is Riemann integrable iff it is Darboux integrable, and the two values coincide. So the two constructions define the same integral, on the same class of functions — which is why one page suffices and why we keep the name Riemann integral while using Darboux sums for proofs (the $\sup$/$\inf$ are cleaner than a net/filter limit over tagged partitions).

Proof. Every tagged sum is trapped, $L(f,P)\le S(f,P,\{t_i\})\le U(f,P)$, since $m_i\le f(t_i)\le M_i$. So whenever the Riemann criterion $U(f,P)-L(f,P)<\epsilon$ holds, all tagged sums for that $P$ lie within $\epsilon$ of the common Darboux value, giving Riemann integrability with the same $I$. Conversely, taking tags where $f$ nearly attains its $\inf$ (resp. $\sup$) on each piece shows tagged sums approximate $L(f,P)$ and $U(f,P)$ arbitrarily well, so Riemann integrability forces $\underline{\int }=>\overline{\int }$. $\square$

We use Darboux as the default below; the Riemann form is the right one for numerical quadrature (midpoint, trapezoid, Simpson) and is the version that generalizes to the gauge (Henstock–Kurzweil) integral, where $\delta$ is allowed to vary with the tag.

3. Classes of Integrable Functions

• Continuous functions are integrable. On the compact interval $[a,b]$, $f$ is uniformly continuous (Heine–Cantor, continuity.md §5), so a fine enough partition makes every $M_i-m_i<\epsilon /(b-a)$, giving $U-L<\epsilon$. This is the most important case.
• Monotone functions are integrable (their oscillation telescopes).
• Bounded with finitely many discontinuities $\Rightarrow$ integrable.
• Lebesgue's criterion (stated): a bounded $f$ is Riemann integrable iff its set of discontinuities has measure zero. This is the sharp characterization; a full proof needs measure theory, which is out of scope for this spine (see the note in README.md).

4. Properties of the Integral

For integrable $f,g$ on $[a,b]$:

• Linearity: $\int (\alpha f+\beta g)=\alpha \int f+\beta \int g$.
• Additivity over intervals: ${\int }_a^{b}f={\int }_a^{c}f+{\int }_c^{b}f$.
• Monotonicity: $f\le g\Rightarrow \int f\le \int g$; in particular $\left|{\int }_a^{b}f\right|\le {\int }_a^b|f|$.
• Products $fg$ and $|f|$ are integrable.
• Uniform limits commute: if $f_n\to f$ uniformly with each $f_n$ integrable, then $f$ is integrable and $\int f_n\to \int f$ (sequences-series.md §7).

Mean Value Theorem for Integrals. If $f$ is continuous on $[a,b]$ there is $\xi \in [a,b]$ with ${\int }_a^{b}f=f(\xi )(b-a)$.

This follows from the IVT applied to the average value $\frac{1}{b-a}{\int }_a^{b}f$, which lies between $\min f$ and $\max f$.

5. The Fundamental Theorem of Calculus

The two halves tie differentiation (differentiation.md) to integration.

FTC, Part I (differentiating an integral). If $f$ is continuous on $[a,b]$ and $F(x)={\int }_a^{x}f(t)dt$, then $F$ is differentiable and $F^{\prime }(x)=f(x)$.

Proof. $\frac{F(x+h)-F(x)}{h}=\frac{1}{h}{\int }_x^{x+h}f=f({\xi }_h)$ for some ${\xi }_h$ between $x$ and $x+h$ (integral MVT); as $h\to 0$, ${\xi }_h\to x$ and continuity gives $f({\xi }_h)\to f(x)$. $\square$

FTC, Part II (evaluating via an antiderivative). If $f$ is integrable and $F$ is any antiderivative ($F^{\prime }=f$) on $[a,b]$, then $${\int }_a^{b}f(x)dx=F(b)-F(a).$$

Proof. For a partition $P$, write $F(b)-F(a)={\sum }_i[F(x_i)-F(x_{i-1})]={\sum }_i{F}^{\prime }({\xi }_i)\Delta x_i={\sum }_{i}f({\xi }_i)\Delta x_i$ by the Mean Value Theorem (differentiation.md §4). This Riemann sum is trapped between $L(f,P)$ and $U(f,P)$, which squeeze to ${\int }_a^{b}f$. $\square$

The FTC is the $n=1$ case of the generalized Stokes theorem. Writing $[a,b]$ as a 1-manifold with boundary $\partial [a,b]=\{b\}-\{a\}$, FTC II reads ${\int }_{[a,b]}dF={\int }_{\partial [a,b]}F$. This is exactly ${\int }_{M}d\omega ={\int }_{\partial M}\omega$ with $\omega =F$ a $0$-form — the punchline assembled in stokes-general.md.

6. Techniques

• Integration by parts: ${\int }_a^{b}u{v}^{\prime }=[uv{]}_a^b-{\int }_a^b{u}^{\prime }v$ — the integral form of the product rule. (The multivariable generalization is the divergence theorem / Green's identities.)
• Change of variables (substitution): if $\phi$ is $C^1$ with ${\phi }^{\prime }$ of one sign, $${\int }_{\phi (a)}^{\phi (b)}f(u)du={\int }_a^{b}f(\phi (x)){\phi }^{\prime }(x)dx.$$ The factor ${\phi }^{\prime }$ is the 1-D Jacobian; in several variables it becomes $|\det D\phi |$ (multiple-integrals.md §4).

7. Improper Integrals

When the interval is unbounded or $f$ is unbounded, define ${\int }_a^{\infty }f={\lim }_{b\to \infty }{\int }_a^{b}f$ and similarly for endpoint singularities, when the limit exists. Such integrals may converge absolutely or only conditionally (e.g. ${\int }_0^{\infty }\frac{\sin x}{x}dx$). The integral test ties their convergence to that of series (sequences-series.md §6.2).

8. Beyond Riemann (pointer)

The Riemann integral suffices for the entire path to Stokes' theorem, but it does not commute well with pointwise limits and cannot integrate badly discontinuous functions. The Lebesgue integral repairs this (dominated/monotone convergence, $L^p$ spaces) at the cost of measure theory. It is deliberately out of scope here; a future measure-theory.md could add it.

9. Where this page is used

• FTC is the prototype the generalized Stokes theorem (stokes-general.md) generalizes.
• Riemann sums / integrability extend to rectangles and Jordan-measurable sets in multiple-integrals.md.
• Change of variables becomes the Jacobian-determinant formula central to multiple integrals and surface integrals (vector-calculus.md).

Next: Metric Spaces and ${\mathbb{R}}^n$ — the multivariable foundations begin.