Unit 09 · Analysis 1A

The Riemann Integral

Area, built rigorously from upper and lower sums. The definition is the careful part; the key theorem — continuous functions are integrable — brings in one genuinely new idea, uniform continuity. Get these right and the Fundamental Theorem (next unit) is a short proof.

By the end you can

Define partitions, upper/lower sums, and the integral.
Prove and apply the integrability criterion $U-L<\varepsilon$.
Explain why continuous and monotone functions are integrable.
Use linearity, additivity, and monotonicity of the integral.

Why it matters & where it comes from

Why it exists

"Area under a curve" needs a precise meaning for wiggly functions. Riemann's squeeze of lower and upper sums supplies it, and tells us exactly which functions can be integrated.

In the real world

Accumulated quantities — distance, work, expected value — are integrals. Probability is the area under a density curve, which is why this unit is the gateway to statistics.

Story & open questions

Archimedes integrated by "exhaustion" in ~250 BC; Riemann gave the modern definition in 1854; Lebesgue generalized it in 1902 into measure theory — the foundation of modern probability.

Partitions and sums

Definition Upper and lower sums

A partition of $[a,b]$ is a finite set $P=\{a=x_0<x_1<\dots<x_n=b\}$. For bounded $f$, with $m_i=\inf f$ and $M_i=\sup f$ on $[x_{i-1},x_i]$ and $\Delta x_i=x_i-x_{i-1}$, $$L(f,P)=\sum_i m_i\,\Delta x_i,\qquad U(f,P)=\sum_i M_i\,\Delta x_i.$$

Lemma Refinement

Adding points to a partition raises $L$ (or keeps it) and lowers $U$ (or keeps it). Consequently every lower sum is $\le$ every upper sum: compare both to their common refinement $L(f,P_1)\le L(f,P_1\cup P_2)\le U(f,P_1\cup P_2)\le U(f,P_2)$.

The definition and the working criterion

Definition Integrable

Bounded $f$ is integrable on $[a,b]$ if $\sup_P L(f,P)=\inf_P U(f,P)$; the common value is $\displaystyle\int_a^b f$.

Theorem Criterion

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

Proof

($\Leftarrow$) $0\le\inf U-\sup L\le U(f,P)-L(f,P) <\varepsilon$ for every $\varepsilon$, so the gap is $0$ (Unit 2, Practice 8).

($\Rightarrow$) If both equal $I$, pick $P_1$ with $L>I-\tfrac\varepsilon2$ and $P_2$ with $U<I+\tfrac\varepsilon2$; their common refinement satisfies $U-L<\varepsilon$. ∎

Example $\int_0^1x\,dx=\tfrac12$, by hand

With $n$ equal pieces, $L=\dfrac{1}{n^2}\sum_{k=0}^{n-1}k=\dfrac{n-1}{2n}$ and $U=\dfrac{1}{n^2}\sum_{k=1}^{n}k=\dfrac{n+1}{2n}$.

$U-L=\tfrac1n\to0$, so $f$ is integrable, and both sums squeeze to $\tfrac12$. ∎

Example The Dirichlet function is not integrable

With $f=1$ on rationals and $0$ on irrationals, every subinterval contains both, so $m_i=0$ and $M_i=1$ always: $L=0$ and $U=b-a$ for every partition. The gap never closes.

Which functions are integrable

Definition Uniform continuity

$f$ is uniformly continuous on a set if for every $\varepsilon>0$ there is $\delta>0$ that works simultaneously at all points: $|x-y|<\delta\Rightarrow|f(x)-f(y)|<\varepsilon$. Fact (proved from completeness): a continuous function on a closed bounded interval $[a,b]$ is automatically uniformly continuous.

Theorem

If $f$ is continuous on $[a,b]$, it is integrable there.

Proof

Given $\varepsilon>0$, uniform continuity provides $\delta>0$ with $|f(x)-f(y)|<\tfrac{\varepsilon}{b-a}$ whenever $|x-y|<\delta$.

Take any partition with all $\Delta x_i<\delta$. On each piece the sup and inf are attained (EVT), at points within $\delta$ of each other, so $M_i-m_i\le\tfrac{\varepsilon}{b-a}$ and $U-L\le\tfrac{\varepsilon}{b-a}\sum\Delta x_i=\varepsilon$. ∎

Theorem Monotone works too

A monotone function on $[a,b]$ is integrable — even with jump discontinuities.

Proof (increasing case)

On the uniform partition into $n$ pieces, $m_i=f(x_{i-1})$ and $M_i=f(x_i)$, so the sum telescopes: $U-L=\dfrac{b-a}{n}\sum_i\big(f(x_i)-f(x_{i-1})\big) =\dfrac{(b-a)\big(f(b)-f(a)\big)}{n}\to0$. ∎

Properties

Theorem Basic rules

For integrable $f,g$: linearity $\int(\alpha f+\beta g)=\alpha\!\int f+\beta\!\int g$; additivity $\int_a^b=\int_a^c+\int_c^b$; monotonicity: $f\le g\Rightarrow\int f\le\int g$; and $\left|\int f\right|\le\int|f|$. Convention: $\int_b^a f=-\int_a^b f$.

Integrability is not antidifferentiation Nothing here mentions antiderivatives. Integrability is about the gap $U-L$ closing. The connection to derivatives is a theorem — the next unit — not a definition.

Common pitfalls

• $m_i,M_i$ are the inf/sup over each subinterval, not endpoint values (unless $f$ is monotone).
• A partition must include both endpoints $a$ and $b$.
• Not every bounded function is integrable — Dirichlet is the standard counterexample.
• One partition with a small gap suffices for the criterion; you never need "all partitions."

Practice

Original problems written for this guide in the style of Spivak's exercises. Write a full solution before opening the answer.

Practice 1 · By hand

Using equal pieces, show $\int_0^1 x^2\,dx=\tfrac13$.

Hint

$U=\dfrac{1}{n^3}\sum_{k=1}^nk^2=\dfrac{(n+1)(2n+1)}{6n^2}\to\tfrac13$, and $L$ (sum to $n-1$) has the same limit.

Practice 2 · Dirichlet

Show the Dirichlet function is not Riemann integrable on $[0,1]$.

Answer

Density of rationals and irrationals gives $m_i=0$, $M_i=1$ on every subinterval, so $L=0$ and $U=1$ for all partitions; the criterion can never be met.

Practice 3 · Bounding

If $1\le f\le4$ on $[0,2]$ and $f$ is integrable, bound $\int_0^2 f$.

Answer

Monotonicity against the constant functions $1$ and $4$: $2\le\int_0^2f\le8$.

Practice 4 · A step function

Fix $c\in(0,1)$ and let $f(x)=0$ for $x<c$, $f(x)=1$ for $x\ge c$. Prove $f$ is integrable on $[0,1]$ and find $\int_0^1f$.

Solution

Trap $c$ inside one subinterval of length $<\varepsilon$; all other subintervals have $M_i=m_i$, so $U-L<\varepsilon$. The value is the length of $[c,1]$: $1-c$.

Practice 5 · Monotone

Write out the proof that an increasing function on $[a,b]$ is integrable.

Solution

Uniform partition, $n$ pieces: $U-L=\tfrac{b-a}{n}\sum(f(x_i)-f(x_{i-1})) =\tfrac{(b-a)(f(b)-f(a))}{n}$, which drops below any $\varepsilon$ for large $n$.

Practice 6 · One bad point

Suppose $f$ is integrable on $[a,b]$ and $g$ equals $f$ except at one point. Prove $g$ is integrable with the same integral.

Solution

$g-f$ is $0$ except at one point $p$; trap $p$ in an interval of length $<\varepsilon/(2|g(p)-f(p)|+1)$ to see $\int(g-f)=0$ by the criterion. Then $g=f+(g-f)$ and linearity finishes it.

Practice 7 · Absolute value

Assuming $|f|$ is integrable when $f$ is, prove $\left|\int_a^b f\right|\le\int_a^b|f|$.

Solution

From $-|f|\le f\le|f|$ and monotonicity, $-\int|f|\le\int f\le\int|f|$ — which is the claim.

Practice 8 · Riemann sums

Using right endpoints on equal pieces, compute $\int_0^2x\,dx$ from the definition.

Solution

$\dfrac2n\sum_{k=1}^n\dfrac{2k}{n}=\dfrac{4}{n^2}\cdot\dfrac{n(n+1)}2 =\dfrac{2(n+1)}{n}\to2$.

Expanded topic checklist

Course coverage. This unit is part of the complete MAT158H1 topic map. It should cover the following items before you move on:

Riemann definition. Use partitions, tags, upper sums, lower sums, and mesh to define the integral.
Integrability criteria. Prove integrability for continuous, monotone, and simple discontinuous examples.
Properties. Use linearity, order, additivity over intervals, and absolute-value inequalities for integrals.
Approximation. Relate sums and integrals through limiting procedures.

Textbook/source alignment

Use the current UofT MAT158H1 calendar topics as the coverage spine, and pair lecture notes with a Spivak-style proof-calculus text for extra reading. The exact assigned textbook should still be checked against the instructor syllabus.

Core theorem map

The Riemann spine is: bounded function, partitions, upper-lower gap, integrability, then algebraic properties of the integral.

Study routine. Every definition proof should end by making an upper-lower gap smaller than an arbitrary epsilon.

Deep practice extensions

Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.

Extension 1 · Criterion

State the Riemann integrability criterion using upper and lower sums.

Guide

A bounded function is integrable iff for every $\epsilon>0$ some partition satisfies $U(f,P)-L(f,P)<\epsilon$.

Extension 2 · Monotone integrability

Sketch why every monotone function on $[a,b]$ is Riemann integrable.

Guide

Use equally spaced partitions. The total upper-lower gap telescopes to roughly $(f(b)-f(a))\Delta x$, which can be made small.

Extension 3 · Order property

Prove if $f\le g$ on $[a,b]$, then $\int f\le\int g$.

Guide

For every partition, lower/upper sums respect the order; pass to integrals, or apply nonnegativity to $g-f$.

Extension 4 · One-point change

Does changing a Riemann integrable function at one point change its integral?

Guide

No. Isolate that point in a tiny interval; the contribution to upper-lower sums can be made arbitrarily small.

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