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Home/Calculus/Chapter 6/The Riemann Integral

CALC-6.1

4-5 hours

The Riemann Integral

Build the definite integral rigorously from Riemann sums and Darboux sums, establishing precise conditions for when a function is integrable.

Learning Objectives

Understand partitions and their refinements

Define Riemann sums and their properties

Master Darboux upper and lower sums

Prove the definition of the Riemann integral

Establish necessary and sufficient conditions for integrability

Identify classes of integrable functions

Apply integrability criteria to specific functions

Understand the Du Bois-Reymond criterion

1. The Area Problem

Motivating Question

How do we compute the area under the curve $y = x^2$ from $x = 0$ to $x = 1$ ?

The ancient Greeks approximated areas using inscribed/circumscribed shapes. We formalize this by dividing $[a, b]$ into subintervals and summing rectangle areas.

2. Partitions and Riemann Sums

Definition 6.1: Partition

Let $f(x)$ be defined on $[a, b]$ . A partition of $[a, b]$ is a finite set of points:

\Delta: a = x_0 < x_1 < x_2 < \cdots < x_{n-1} < x_n = b

This divides $[a, b]$ into $n$ subintervals $[x_{k-1}, x_k]$ for $k = 1, 2, \ldots, n$ .

Definition 6.2: Norm of Partition

The norm (or mesh) of partition $\Delta$ is:

\|\Delta\| = \max\{\Delta x_1, \Delta x_2, \ldots, \Delta x_n\}

where $\Delta x_k = x_k - x_{k-1}$ is the length of the $k$ -th subinterval.

Definition 6.3: Riemann Sum

Given partition $\Delta$ and sample points $\xi_k \in [x_{k-1}, x_k]$ , the Riemann sum is:

S_\Delta(f, \xi) = \sum_{k=1}^{n} f(\xi_k) \Delta x_k = \sum_{k=1}^{n} f(\xi_k)(x_k - x_{k-1})

This represents the sum of rectangle areas with heights $f(\xi_k)$ and widths $\Delta x_k$ .

Example 6.1: Computing a Riemann Sum

Problem: Compute the Riemann sum for $f(x) = x^2$ on $[0, 1]$ with uniform partition $n = 4$ and right endpoints.

Solution:

Partition: $x_k = k/4$ for $k = 0, 1, 2, 3, 4$ . So $\Delta x_k = 1/4$ .

Right endpoints: $\xi_k = k/4$ .

S_4 = \sum_{k=1}^{4} \left(\frac{k}{4}\right)^2 \cdot \frac{1}{4} = \frac{1}{64}(1 + 4 + 9 + 16) = \frac{30}{64} = \frac{15}{32}

Note: The exact value is $\int_0^1 x^2\,dx = 1/3 \approx 0.333$ , while $15/32 \approx 0.469$ .

3. Darboux Upper and Lower Sums

Definition 6.4: Darboux Sums

For partition $\Delta$ of $[a, b]$ and bounded function $f$ , define:

M_k = \sup_{x \in [x_{k-1}, x_k]} f(x), \quad m_k = \inf_{x \in [x_{k-1}, x_k]} f(x)

The upper Darboux sum and lower Darboux sum are:

\overline{S}_\Delta = \sum_{k=1}^{n} M_k \Delta x_k

Upper sum

\underline{S}_\Delta = \sum_{k=1}^{n} m_k \Delta x_k

Lower sum

Theorem 6.1: Bounds on Riemann Sums

For any partition $\Delta$ and sample points $\{\xi\}$ :

\underline{S}_\Delta \leq S_\Delta(f, \xi) \leq \overline{S}_\Delta

Proof of Theorem 6.1:

Since $m_k \leq f(\xi_k) \leq M_k$ for each $k$ , multiplying by $\Delta x_k > 0$ and summing:

\sum_{k=1}^n m_k \Delta x_k \leq \sum_{k=1}^n f(\xi_k) \Delta x_k \leq \sum_{k=1}^n M_k \Delta x_k

∎

4. Refinement of Partitions

Definition 6.5: Refinement

Partition $\Delta_2$ is a refinement of $\Delta_1$ (written $\Delta_1 \subset \Delta_2$ ) if every partition point of $\Delta_1$ is also a partition point of $\Delta_2$ .

The common refinement of $\Delta_1$ and $\Delta_2$ is $\Delta^* = \Delta_1 \cup \Delta_2$ .

Theorem 6.2: Refinement Lemma

If $\Delta_2 \supset \Delta_1$ and $\Delta_2$ has $k$ more points than $\Delta_1$ , then:

0 \leq \overline{S}_{\Delta_1} - \overline{S}_{\Delta_2} \leq k\|\Delta_1\|(M - m)

0 \leq \underline{S}_{\Delta_2} - \underline{S}_{\Delta_1} \leq k\|\Delta_1\|(M - m)

where $M = \sup f$ , $m = \inf f$ on $[a, b]$ .

Corollary 6.1: Upper and Lower Sum Comparison

For any two partitions $\Delta_1, \Delta_2$ of $[a, b]$ :

\underline{S}_{\Delta_1} \leq \overline{S}_{\Delta_2}

Every lower sum is ≤ every upper sum (even for different partitions).

5. Upper and Lower Integrals

Definition 6.6: Darboux Integrals

For bounded $f$ on $[a, b]$ , define the upper integral and lower integral:

\overline{\int_a^b} f(x)\,dx = \inf_\Delta \overline{S}_\Delta

Upper (Darboux) integral

\underline{\int_a^b} f(x)\,dx = \sup_\Delta \underline{S}_\Delta

Lower (Darboux) integral

Theorem 6.3: Darboux's Theorem

For bounded $f$ on $[a, b]$ : $\forall \epsilon > 0$ , $\exists \delta > 0$ such that for any partition $\Delta$ with $\|\Delta\| < \delta$ :

\overline{\int_a^b} f - \epsilon < \overline{S}_\Delta < \overline{\int_a^b} f + \epsilon

\underline{\int_a^b} f - \epsilon < \underline{S}_\Delta < \underline{\int_a^b} f + \epsilon

Remark 6.1

The upper and lower integrals always exist for bounded functions, and we always have:

\underline{\int_a^b} f \leq \overline{\int_a^b} f

6. Definition of the Riemann Integral

Definition 6.7: Riemann Integrability

Let $f$ be bounded on $[a, b]$ . If there exists $A \in \mathbb{R}$ such that:

\forall \epsilon > 0, \exists \delta > 0: \|\Delta\| < \delta \Rightarrow |S_\Delta(f, \xi) - A| < \epsilon

for any partition $\Delta$ and any sample points $\{\xi\}$ , then $f$ is Riemann integrable on $[a, b]$ .

We write $f \in R[a, b]$ and define:

\int_a^b f(x)\,dx = A = \lim_{\|\Delta\| \to 0} \sum_{k=1}^n f(\xi_k) \Delta x_k

Theorem 6.4: Uniqueness

If $f \in R[a, b]$ , then the integral value $A$ is unique.

Theorem 6.5: Necessary Condition

If $f \in R[a, b]$ , then $f$ is bounded on $[a, b]$ .

Proof of Theorem 6.5:

Suppose $f$ is unbounded on $[a, b]$ . For any partition $\Delta$ , $f$ is unbounded on at least one subinterval $[x_{k-1}, x_k]$ . By choosing sample points appropriately, we can make $|S_\Delta(f, \xi)|$ arbitrarily large, contradicting convergence to a finite $A$ .

∎

7. Integrability Criteria

Theorem 6.6: Darboux Criterion

For bounded $f$ on $[a, b]$ :

f \in R[a, b] \Leftrightarrow \overline{\int_a^b} f = \underline{\int_a^b} f

Theorem 6.7: Oscillation Criterion

Define the oscillation on $[x_{k-1}, x_k]$ as $\omega_k = M_k - m_k$ . Then:

f \in R[a, b] \Leftrightarrow \forall \epsilon > 0, \exists \delta > 0: \|\Delta\| < \delta \Rightarrow \sum_{k=1}^n \omega_k \Delta x_k < \epsilon

Equivalently: $\overline{S}_\Delta - \underline{S}_\Delta < \epsilon$ .

Theorem 6.8: Du Bois-Reymond Criterion

For bounded $f$ on $[a, b]$ :

f \in R[a, b] \Leftrightarrow \forall \epsilon > 0, \forall \sigma > 0, \exists \text{ partition } \Delta:

\sum_{k: \omega_k \geq \epsilon} \Delta x_k < \sigma

The total length of “bad” subintervals (where oscillation ≥ ε) can be made arbitrarily small.

8. Classes of Integrable Functions

Example 6.2: Continuous Functions

Theorem: If $f \in C[a, b]$ , then $f \in R[a, b]$ .

Proof Sketch:

Continuous on closed interval ⟹ uniformly continuous.

Given $\epsilon > 0$ , choose $\delta$ from uniform continuity.

For $\|\Delta\| < \delta$ : $\omega_k < \epsilon/(b-a)$ on each subinterval, so $\sum \omega_k \Delta x_k < \epsilon$ .

Example 6.3: Monotonic Functions

Theorem: If $f$ is monotonic on $[a, b]$ , then $f \in R[a, b]$ .

Proof Sketch:

For increasing $f$ : $M_k = f(x_k)$ , $m_k = f(x_{k-1})$ .

\overline{S} - \underline{S} = \sum_{k=1}^n [f(x_k) - f(x_{k-1})]\Delta x_k \leq \|\Delta\|[f(b) - f(a)]

This can be made $< \epsilon$ by choosing $\|\Delta\|$ small.

Example 6.4: Bounded Functions with Finitely Many Discontinuities

Theorem: If $f$ is bounded on $[a, b]$ and has only finitely many discontinuities, then $f \in R[a, b]$ .

Proof Idea:

Enclose each discontinuity in a small interval. The total contribution from these intervals can be bounded by $2M \cdot k\delta$ where $k$ is the number of discontinuities.

Example 6.5: The Riemann Function (Thomae's Function)

Definition:

R(x) = \begin{cases} \frac{1}{q} & x = \frac{p}{q} \text{ in lowest terms}, q > 0 \\ 0 & x \text{ irrational} \end{cases}

Key Facts:

$R$ is discontinuous at every rational
$R$ is continuous at every irrational
$R \in R[0, 1]$ with $\int_0^1 R(x)\,dx = 0$

Example 6.6: Dirichlet Function (NOT Integrable)

Definition:

D(x) = \mathbf{1}_\mathbb{Q}(x) = \begin{cases} 1 & x \in \mathbb{Q} \\ 0 & x \notin \mathbb{Q} \end{cases}

Why NOT Integrable:

For any partition: $M_k = 1$ , $m_k = 0$ on every subinterval.

Thus $\overline{S} = b - a$ , $\underline{S} = 0$ for all partitions, so $\overline{\int} \neq \underline{\int}$ .

9. Basic Conventions

Remark 6.2: Notation Conventions

Integration variable: The definite integral is independent of the variable name:

\int_a^b f(x)\,dx = \int_a^b f(t)\,dt = \int_a^b f(u)\,du

Reversed limits:

\int_b^a f(x)\,dx = -\int_a^b f(x)\,dx

Equal limits:

\int_a^a f(x)\,dx = 0

10. Common Mistakes

❌ Bounded ⟹ Integrable

The Dirichlet function is bounded but not integrable!

❌ Discontinuous ⟹ Not Integrable

Functions with “few” discontinuities (measure zero) can be integrable.

❌ Confusing Riemann and Darboux sums

Riemann sums use sample points; Darboux uses sup/inf.

❌ Forgetting ||Δ|| → 0 means ALL partitions

The limit must work for any sequence of partitions with shrinking norm.

11. Additional Examples

Example 6.7: Computing ∫₀¹ x² Using Definition

Problem: Compute $\int_0^1 x^2\,dx$ directly from the definition using Riemann sums.

Solution:

Use uniform partition with right endpoints: $x_k = k/n$ , $\Delta x_k = 1/n$ .

S_n = \sum_{k=1}^{n} \left(\frac{k}{n}\right)^2 \cdot \frac{1}{n} = \frac{1}{n^3} \sum_{k=1}^{n} k^2

Using the formula $\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$ :

S_n = \frac{1}{n^3} \cdot \frac{n(n+1)(2n+1)}{6} = \frac{(n+1)(2n+1)}{6n^2}

Taking the limit:

\lim_{n \to \infty} S_n = \lim_{n \to \infty} \frac{(1+1/n)(2+1/n)}{6} = \frac{1 \cdot 2}{6} = \frac{1}{3}

Example 6.8: Computing ∫₀¹ eˣ dx

Problem: Compute $\int_0^1 e^x\,dx$ using Riemann sums.

Solution:

With right endpoints: $\xi_k = k/n$ .

S_n = \sum_{k=1}^{n} e^{k/n} \cdot \frac{1}{n} = \frac{1}{n} \sum_{k=1}^{n} (e^{1/n})^k

This is a geometric series with ratio $r = e^{1/n}$ :

S_n = \frac{1}{n} \cdot \frac{e^{1/n}(e - 1)}{e^{1/n} - 1} = \frac{e^{1/n}(e-1)}{n(e^{1/n}-1)}

Since $\lim_{n \to \infty} n(e^{1/n} - 1) = 1$ (by L'Hôpital or Taylor):

\int_0^1 e^x\,dx = \lim_{n \to \infty} S_n = e - 1

Example 6.9: Upper and Lower Sums for Step Function

Problem: Let $f(x) = \lfloor x \rfloor$ on $[0, 3]$ . Show $f \in R[0, 3]$ and find the integral.

Solution:

$f(x) = 0$ on $[0, 1)$ , $f(x) = 1$ on $[1, 2)$ , $f(x) = 2$ on $[2, 3]$ .

Use partition $\Delta = \{0, 1, 2, 3\}$ :

On $[0, 1]$ : $M_1 = 1, m_1 = 0$
On $[1, 2]$ : $M_2 = 2, m_2 = 1$
On $[2, 3]$ : $M_3 = 2, m_3 = 2$

$\overline{S} - \underline{S} = (1-0) + (2-1) + (2-2) = 2$ .

With finer partitions avoiding integers: $\Delta' = \{0, 1-\epsilon, 1+\epsilon, 2-\epsilon, 2+\epsilon, 3\}$ :

\overline{S}' - \underline{S}' \leq 4\epsilon \to 0

So $f \in R[0, 3]$ with $\int_0^3 f = 0 \cdot 1 + 1 \cdot 1 + 2 \cdot 1 = 3$ .

Example 6.10: Verifying Integrability via Oscillation

Problem: Show that $f(x) = \sin(1/x)$ for $x \in (0, 1]$ , $f(0) = 0$ is integrable on $[0, 1]$ .

Solution:

$f$ is continuous on $(0, 1]$ and bounded by 1.

For any $\epsilon > 0$ , choose $\delta = \epsilon/4$ .

Split $[0, 1] = [0, \delta] \cup [\delta, 1]$ :

On $[0, \delta]$ : oscillation ≤ 2, contribution to sum ≤ $2\delta = \epsilon/2$
On $[\delta, 1]$ : $f$ is uniformly continuous, can make oscillation sum $< \epsilon/2$

Total $\sum \omega_k \Delta x_k < \epsilon$ , so $f \in R[0, 1]$ .

Example 6.11: Showing Riemann Sums Converge

Problem: Prove $\lim_{n \to \infty} \frac{1}{n}\sum_{k=1}^n \frac{1}{1 + (k/n)^2} = \frac{\pi}{4}$ .

Solution:

Recognize this as a Riemann sum for $f(x) = \frac{1}{1+x^2}$ on $[0, 1]$ :

\frac{1}{n}\sum_{k=1}^n f(k/n) = \sum_{k=1}^n f(\xi_k) \Delta x_k \to \int_0^1 \frac{1}{1+x^2}\,dx

Computing the integral:

\int_0^1 \frac{1}{1+x^2}\,dx = \arctan x \Big|_0^1 = \arctan 1 - \arctan 0 = \frac{\pi}{4}

12. Practice Problems

Problem Set A: Riemann Sums

1.Compute $\int_0^2 x^3\,dx$ using Riemann sums with right endpoints.
2.Show $\lim_{n \to \infty} \frac{1}{n}\sum_{k=1}^n \sqrt{k/n} = 2/3$ .
3.For $f(x) = x$ on $[0, 1]$ , find $\overline{S}_\Delta$ and $\underline{S}_\Delta$ for uniform partition with $n$ points.
4.Evaluate $\lim_{n \to \infty} \frac{1^p + 2^p + \cdots + n^p}{n^{p+1}}$ using Riemann sums.

Problem Set B: Integrability

5.Prove that if $f$ is bounded and has countably many discontinuities, then $f \in R[a, b]$ .
6.Show that $f(x) = x \cdot D(x)$ (where $D$ is Dirichlet) is integrable on $[0, 1]$ .
7.If $f \in R[a, b]$ and $g$ differs from $f$ at finitely many points, show $g \in R[a, b]$ .
8.Give an example of a function integrable on $[0, 1]$ that is discontinuous at every rational.

Problem Set C: Darboux Sums

9.Prove: $\overline{S}_{\Delta_1 \cup \Delta_2} \leq \min(\overline{S}_{\Delta_1}, \overline{S}_{\Delta_2})$ .
10.Show that $\overline{\int} f + \overline{\int} g \geq \overline{\int}(f + g)$ with an example where equality fails.
11.For $f(x) = x^2$ on $[0, 2]$ , compute $\overline{S}$ and $\underline{S}$ for partition $\{0, 1, 2\}$ .
12.Prove: If $f \leq g$ on $[a, b]$ , then $\overline{\int} f \leq \overline{\int} g$ .

Remark 6.3: Answers to Selected Problems

Problem 1: $\int_0^2 x^3\,dx = 4$

Problem 4: $\frac{1}{p+1}$

Problem 8: Riemann function $R(x)$

Problem 11: $\overline{S} = 5, \underline{S} = 1$

13. Historical Context

Bernhard Riemann (1826-1866)

German mathematician who formalized the definition of the integral in his 1854 habilitation lecture. He introduced the concept of using arbitrary partitions and sample points, asking when the limit exists independent of these choices.

Jean-Gaston Darboux (1842-1917)

French mathematician who developed the upper and lower sum approach (1875), providing an equivalent but often more convenient criterion for integrability. This approach eliminates dependence on sample points.

Paul du Bois-Reymond (1831-1889)

German mathematician who gave a precise characterization of integrable functions in terms of their oscillation on “bad” subintervals, leading to measure-theoretic interpretations.

Henri Lebesgue (1875-1941)

French mathematician who extended integration theory (1902) to handle more functions. His measure-theoretic approach showed that Riemann integrable functions are exactly those continuous “almost everywhere.”

Remark 6.4: The Evolution of Integration

Integration concepts evolved over centuries:

Archimedes (c. 287-212 BC): Method of exhaustion for areas
Newton & Leibniz (1660s): Infinitesimal calculus, antiderivatives
Cauchy (1823): First rigorous definition for continuous functions
Riemann (1854): General definition allowing discontinuities
Lebesgue (1902): Measure-theoretic extension

14. Connections to Other Topics

Measure Theory

Riemann integrability is characterized by the set of discontinuities having measure zero. This connects to Lebesgue's criterion and motivates the Lebesgue integral.

Numerical Analysis

Riemann sums form the basis of numerical integration methods: left/right rectangle rules, midpoint rule, and their generalizations (trapezoidal, Simpson's rule).

Probability Theory

The integral of a PDF over an interval gives probability. The definition of expected value $E[X] = \int x f(x)\,dx$ relies on Riemann/Lebesgue integration.

Physics

Work, center of mass, moments of inertia—all computed as integrals. The Riemann sum interpretation gives physical meaning to these quantities.

15. Summary of Key Results

Concept	Formula/Statement
Riemann Sum	$S_\Delta = \sum_{k=1}^n f(\xi_k)\Delta x_k$
Upper Sum	$\overline{S}_\Delta = \sum_{k=1}^n M_k \Delta x_k$
Lower Sum	$\underline{S}_\Delta = \sum_{k=1}^n m_k \Delta x_k$
Integrability	$f \in R[a,b] \Leftrightarrow \overline{\int} f = \underline{\int} f$
Oscillation Criterion	$\forall \epsilon > 0, \exists \Delta: \sum \omega_k \Delta x_k < \epsilon$
Continuous ⟹ Integrable	$f \in C[a,b] \Rightarrow f \in R[a,b]$
Monotone ⟹ Integrable	$f$ monotonic on $[a,b] \Rightarrow f \in R[a,b]$

16. Advanced Topics Preview

Lebesgue's Criterion

A bounded function $f$ on $[a, b]$ is Riemann integrable if and only if the set of discontinuities of $f$ has Lebesgue measure zero.

This explains why the Riemann function is integrable (discontinuous only at rationals, a set of measure zero) while the Dirichlet function is not (discontinuous everywhere).

Improper Integrals

When the interval is unbounded or the function is unbounded, we define improper integrals as limits:

\int_a^\infty f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx

\int_a^b \frac{1}{(x-a)^p}\,dx = \lim_{\epsilon \to 0^+} \int_{a+\epsilon}^b \frac{1}{(x-a)^p}\,dx

Riemann-Stieltjes Integral

A generalization where we integrate with respect to a function $g$ rather than $x$ :

\int_a^b f(x)\,dg(x) = \lim_{\|\Delta\| \to 0} \sum_{k=1}^n f(\xi_k)[g(x_k) - g(x_{k-1})]

This is useful in probability (expectations), physics (moments), and functional analysis.

Multiple Integrals

The Riemann integral extends to higher dimensions. For a function on a rectangle $R = [a,b] \times [c,d]$ :

\iint_R f(x,y)\,dA = \lim_{\|\Delta\| \to 0} \sum_{i,j} f(\xi_{ij}, \eta_{ij}) \Delta x_i \Delta y_j

Fubini's theorem connects double integrals to iterated single integrals.

17. Computational Tips

Recognizing Riemann Sums

To evaluate $\lim_{n \to \infty} \frac{1}{n}\sum_{k=1}^n f(k/n)$ :

Identify this as a Riemann sum on $[0, 1]$
The limit equals $\int_0^1 f(x)\,dx$
Generalize: $\frac{1}{n}\sum f(a + k(b-a)/n) \to \int_a^b f(x)\,dx \cdot \frac{1}{b-a}$

Power Sum Formulas

Useful for computing Riemann sums:

$\sum_{k=1}^n k = \frac{n(n+1)}{2}$

$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$

$\sum_{k=1}^n k^3 = \left[\frac{n(n+1)}{2}\right]^2$

Proving Integrability

Strategy checklist:

Check if $f$ is continuous → done!
Check if $f$ is monotonic → done!
Check if discontinuities are “sparse”
Use oscillation criterion directly

Showing Non-Integrability

To show $f \notin R[a,b]$ :

Find partitions where $\overline{S} - \underline{S}$ is bounded away from 0
Show $M_k - m_k$ is constant on every subinterval
Example: Dirichlet function has $M_k = 1, m_k = 0$ always

18. Study Guide

Key Definitions to Memorize

Partition, norm of partition
Riemann sum with sample points
Upper/lower Darboux sums
Upper/lower Darboux integrals
Oscillation $\omega_k = M_k - m_k$

Key Theorems to Understand

Refinement decreases upper sums, increases lower sums
Darboux criterion: $\overline{\int} = \underline{\int} \Leftrightarrow f \in R[a,b]$
Oscillation criterion: $\sum \omega_k \Delta x_k \to 0$
Continuous functions are integrable
Monotonic functions are integrable

Important Examples

Computing $\int x^n$ from definition
Dirichlet function (not integrable)
Riemann/Thomae function (integrable, discontinuous at rationals)
Step functions (integrable)

Common Exam Questions

Evaluate limits using Riemann sum interpretation
Prove a specific function is/isn't integrable
Compute upper/lower sums for a given partition
State and prove the oscillation criterion

19. What's Next?

In the next section, we explore the properties of definite integrals:

Linearity:

\int(\alpha f + \beta g) = \alpha\int f + \beta\int g

Interval additivity:

\int_a^b = \int_a^c + \int_c^b

Comparison theorems and estimation

Absolute value and product integrability

Riemann Integral Quiz

Questions

Correct

Accuracy

What is a partition of

[a,b]

Easy

Not attempted

The Riemann sum

S_\Delta(f, \xi)

equals:

Easy

Not attempted

The Darboux upper sum

\overline{S}_\Delta

uses:

Easy

Not attempted

For any Riemann sum, we have:

Medium

Not attempted

\Delta_2

is a refinement of

\Delta_1

, then:

Medium

Not attempted

A bounded function

f

is Riemann integrable on

[a,b]

iff:

Medium

Not attempted

Which function is NOT Riemann integrable on

[0,1]

Hard

Not attempted

A continuous function on

[a,b]

is:

Easy

Not attempted

A monotonic function on

[a,b]

is:

Medium

Not attempted

The Du Bois-Reymond criterion states

f \in R[a,b]

iff:

Hard

Not attempted

Frequently Asked Questions

What is the intuition behind Riemann sums?

Riemann sums approximate the area under a curve by dividing it into rectangles. As the partition gets finer (||Δ|| → 0), these approximations converge to the true area if the function is integrable.

Why do we need Darboux sums?

Darboux sums provide upper and lower bounds independent of sample point choice. They make proving integrability easier since we only need to show the upper and lower integrals are equal.

What's the difference between Riemann and Darboux integrals?

They are equivalent for bounded functions! A function is Riemann integrable iff it's Darboux integrable, and the integral values are the same.

Why isn't the Dirichlet function integrable?

For any partition, every subinterval contains both rationals and irrationals, so M_k = 1, m_k = 0. Thus S̄ = 1, S = 0 for all partitions.

How does the Riemann function (Thomae's function) differ?

The Riemann function R(x) = 1/q for x = p/q in lowest terms, R(x) = 0 for irrational x. It's discontinuous at rationals but integrable because rationals are 'sparse'.

What does 'norm of partition' ||Δ|| mean?

||Δ|| = max{Δx₁, Δx₂, ..., Δxₙ} is the length of the longest subinterval. As ||Δ|| → 0, the partition becomes finer.

Can a function with infinitely many discontinuities be integrable?

Yes! The key is that discontinuities must form a 'small' set (measure zero). The Riemann function has countably many discontinuities but is integrable.

What's the relationship between uniform continuity and integrability?

Uniform continuity on [a,b] implies integrability. Given ε, uniform continuity provides δ such that ||Δ|| < δ ensures S̄ - S < ε.

Key Takeaways

Riemann Sum

S_\Delta = \sum f(\xi_k)\Delta x_k

Darboux Bounds

\underline{S} \leq S_\Delta \leq \overline{S}

Integrability

$f \in R[a,b] \Leftrightarrow \overline{\int} = \underline{\int}$

Key Classes

Continuous, monotonic functions are integrable

Refinement Property

\Delta_2 \supset \Delta_1 \Rightarrow \overline{S}_{\Delta_2} \leq \overline{S}_{\Delta_1}

Oscillation Criterion

\sum \omega_k \Delta x_k \to 0

Quick Reference: Integrability Tests

✓ Always Integrable

Continuous functions
Monotonic functions
Bounded + finite discontinuities

? May Be Integrable

Bounded + countable disc.
Riemann/Thomae function
Piecewise continuous

✗ Not Integrable

Dirichlet function
Unbounded functions
Dense discontinuities

Key Definitions Checklist

Partition:

a = x_0 < x_1 < \cdots < x_n = b

Norm:

\|\Delta\| = \max \Delta x_k

Upper sum:

\overline{S} = \sum M_k \Delta x_k

Lower sum:

\underline{S} = \sum m_k \Delta x_k

Oscillation:

\omega_k = M_k - m_k

Integral:

\lim_{\|\Delta\| \to 0} S_\Delta

The Riemann Integral

1. The Area Problem

Motivating Question

2. Partitions and Riemann Sums

3. Darboux Upper and Lower Sums

4. Refinement of Partitions

5. Upper and Lower Integrals

6. Definition of the Riemann Integral

7. Integrability Criteria

8. Classes of Integrable Functions

9. Basic Conventions

10. Common Mistakes

❌ Bounded ⟹ Integrable

❌ Discontinuous ⟹ Not Integrable

❌ Confusing Riemann and Darboux sums

❌ Forgetting ||Δ|| → 0 means ALL partitions

11. Additional Examples

12. Practice Problems

Problem Set A: Riemann Sums

Problem Set B: Integrability

Problem Set C: Darboux Sums

13. Historical Context

Bernhard Riemann (1826-1866)

Jean-Gaston Darboux (1842-1917)

Paul du Bois-Reymond (1831-1889)

Henri Lebesgue (1875-1941)

14. Connections to Other Topics

Measure Theory

Numerical Analysis

Probability Theory

Physics

15. Summary of Key Results

16. Advanced Topics Preview

Lebesgue's Criterion

Improper Integrals

Riemann-Stieltjes Integral

Multiple Integrals

17. Computational Tips

Recognizing Riemann Sums

Power Sum Formulas

Proving Integrability

Showing Non-Integrability

18. Study Guide

Key Definitions to Memorize

Key Theorems to Understand

Important Examples

Common Exam Questions

19. What's Next?

Frequently Asked Questions

What is the intuition behind Riemann sums?

Why do we need Darboux sums?

What's the difference between Riemann and Darboux integrals?

Why isn't the Dirichlet function integrable?

How does the Riemann function (Thomae's function) differ?

What does 'norm of partition' ||Δ|| mean?

Can a function with infinitely many discontinuities be integrable?

What's the relationship between uniform continuity and integrability?

Riemann Sum

Darboux Bounds

Integrability

Key Classes

Refinement Property

Oscillation Criterion

Quick Reference: Integrability Tests

Key Definitions Checklist

Quick Decision Guide

When Computing Riemann Sums

When Proving Integrability