Integration begins as a limit of finite accumulations. Antiderivatives enter later, through a theorem.
BTT already has school and A-Math routes for definite integrals, area and motion, including Definite Integrals, Area, Accumulation and Motion and Why Integration Feels Like Differentiation Backwards. This guide owns the rigorous real-analysis layer underneath those methods.
Its central distinction is simple but important: the Riemann integral is defined by controlled sums; the Fundamental Theorem of Calculus later proves that for sufficiently regular functions this accumulation can be evaluated through antiderivatives.
Level: university-entry / undergraduate real-analysis enrichment. Prerequisites: limits, continuity, derivatives and basic inequalities.
Reading route: partitions → Riemann sums → upper/lower sums → integrability → continuous/monotone integrability → signed accumulation → FTC I → FTC II → substitution → integration by parts → estimates → improper integrals → convergence tests → counterexamples → practice → solutions.
1. A partition breaks an interval into controlled pieces
A partition P of [a,b] is a finite ordered set
a=x₀<x₁<⋯<x_n=b.
The width of the i-th subinterval is Δx_i=x_i−x_{i−1}.
The mesh of the partition is max_i Δx_i. Sending the mesh to zero means every subinterval becomes uniformly fine.
2. A tagged Riemann sum approximates accumulated quantity
Choose a sample point ξ_i∈[x_{i−1},x_i] in each subinterval. The tagged Riemann sum is
Σ f(ξ_i)Δx_i.
If f is a rate, f(ξ_i)Δx_i approximates the amount accumulated over that small interval.
If f is height, the term is the signed area of a rectangle.
The integral asks whether all sufficiently fine tagged sums are forced toward one number independent of the detailed partition and tag choices.
3. Riemann integrability is a stability statement about all fine sums
A bounded function f on [a,b] is Riemann integrable with integral I if for every ε>0 there exists δ>0 such that every tagged partition with mesh less than δ satisfies
|Σf(ξ_i)Δx_i−I|<ε.
This formulation makes integration another limit-control theorem. The sum is allowed to vary, but sufficiently fine representations all collapse onto the same accumulated value.
4. Upper and lower sums remove tag dependence
For each subinterval I_i=[x_{i−1},x_i], let
m_i=inf_{I_i}f and M_i=sup_{I_i}f.
The lower and upper sums are
L(f,P)=Σm_iΔx_i, U(f,P)=ΣM_iΔx_i.
Every tagged Riemann sum lies between them.
Refining the partition raises lower sums and lowers upper sums, squeezing the possible accumulated values.
5. Darboux’s criterion makes integrability quantitative
A bounded f on [a,b] is Riemann integrable if and only if for every ε>0 there exists a partition P such that
U(f,P)−L(f,P)<ε.
The total unresolved oscillation can be made arbitrarily small.
This criterion is often more practical for proving integrability than controlling every tagged sum directly.
6. Continuous functions on closed intervals are Riemann integrable
If f is continuous on [a,b], compactness implies uniform continuity.
Therefore, for sufficiently short subintervals, the oscillation M_i−m_i is uniformly small.
Then U(f,P)−L(f,P)=Σ(M_i−m_i)Δx_i can be made arbitrarily small.
So continuity on a compact interval gives Riemann integrability.
7. Monotone functions are also Riemann integrable
A bounded monotone function on [a,b] is Riemann integrable even if it has jump discontinuities.
For an increasing function on an equal partition, the upper-minus-lower sum telescopes into a bound proportional to the mesh:
U−L≤mesh(P)[f(b)−f(a)].
Making the mesh small forces the gap to zero.
8. Worked integral from Riemann sums: ∫₀¹x dx
Use n equal subintervals and right endpoints i/n.
The Riemann sum is
Σ_{i=1}^n (i/n)(1/n)=1/n²·n(n+1)/2=(n+1)/(2n).
As n→∞, this tends to 1/2.
Thus ∫₀¹x dx=1/2 directly from the defining accumulation limit.
9. The definite integral is signed accumulation
If f is positive, the integral agrees with geometric area under standard graph assumptions.
If f is negative, the integral contributes negatively.
Therefore ∫_a^b f is net signed accumulation, not automatically total geometric area.
To compute total area between a graph and the x-axis, one integrates |f| or splits at sign changes.
10. Basic integral properties follow from sums
For Riemann-integrable f and g:
- Linearity: ∫(αf+βg)=α∫f+β∫g.
- Additivity: ∫_a^c f+∫_c^b f=∫_a^b f.
- Order: if f≤g, then ∫f≤∫g.
- Absolute estimate: |∫f|≤∫|f| whenever |f| is integrable.
These are not symbolic conventions; they descend from corresponding finite-sum properties.
11. The average value theorem is an integral Mean Value Theorem
If f is continuous on [a,b], then there exists c∈[a,b] such that
∫_a^b f(x)dx=f(c)(b−a).
Equivalently, the average value
(1/(b−a))∫_a^b f
is actually attained somewhere because continuity traps the average between the minimum and maximum values.
12. FTC Part I differentiates accumulated area
Let f be continuous on [a,b] and define
F(x)=∫_a^x f(t)dt.
Then F is differentiable on (a,b) and
F'(x)=f(x).
Accumulating a continuous rate and then differentiating recovers the original rate.
13. Why FTC I is locally plausible
Consider
[F(x+h)−F(x)]/h=(1/h)∫_x^{x+h}f(t)dt.
This is the average value of f over a short interval.
As h→0, continuity forces that local average to approach f(x).
The derivative of accumulated quantity is therefore the instantaneous density being accumulated.
14. FTC Part II evaluates accumulation through any antiderivative
If f is continuous on [a,b] and G is any antiderivative of f on [a,b], then
∫_a^b f(x)dx=G(b)−G(a).
This is the theorem behind the familiar evaluation rule.
The integral was defined by sums; FTC proves that antiderivatives compute those sums efficiently.
15. Worked FTC example: accumulated square
Let F(x)=∫₀^x t²dt.
FTC I gives F'(x)=x².
An antiderivative of t² is t³/3, so FTC II gives
F(x)=x³/3.
The derivative and integral relationship is now theorem-backed in both directions.
16. Substitution is change of variables plus the chain rule
For a suitable differentiable change of variable u=φ(x), the substitution formula is
∫_a^b f(φ(x))φ'(x)dx=∫_{φ(a)}^{φ(b)}f(u)du
under the standard hypotheses ensuring the change of variables is legitimate.
It is not a literal cancellation of “du” and “dx”. The notation reflects a theorem derived from the chain rule and FTC.
17. Integration by parts is the product rule integrated
Since (uv)’=u’v+uv’, integrating gives
∫_a^b u(x)v'(x)dx=[u(x)v(x)]_a^b−∫_a^b u'(x)v(x)dx.
The method transfers a derivative from one factor to the other.
Its usefulness depends on choosing which factor becomes simpler after differentiation and which has a manageable antiderivative.
18. Integral estimates can be more useful than exact values
If m≤f(x)≤M on [a,b], then
m(b−a)≤∫_a^b f≤M(b−a).
If |f|≤M, then |∫f|≤M(b−a).
These estimates support error bounds, convergence arguments and stability proofs even when no elementary antiderivative exists.
19. The Dirichlet function is not Riemann integrable
Define f(x)=1 on rational numbers and f(x)=0 on irrational numbers in [0,1].
Every interval contains both rationals and irrationals.
Therefore every subinterval has infimum 0 and supremum 1.
For every partition, L(f,P)=0 and U(f,P)=1.
The gap never shrinks, so the function is not Riemann integrable.
20. A bounded function can be discontinuous and still integrable
The step function with one jump is Riemann integrable.
More generally, bounded functions with sufficiently limited discontinuity can be Riemann integrable.
Lebesgue’s criterion states that a bounded function on [a,b] is Riemann integrable exactly when its set of discontinuities has measure zero.
The measure language belongs to R25.07; here it explains why “discontinuous” and “nonintegrable” are not synonyms.
21. Improper integrals are defined by limits, not by ordinary Riemann integration
If the interval is unbounded, define for example
∫_a^∞ f(x)dx=lim_{R→∞}∫_a^R f(x)dx
when the limit exists finitely.
If f is unbounded near an endpoint b, define
∫_a^b f(x)dx=lim_{t→b−}∫_a^t f(x)dx
when that finite limit exists.
These are new limit definitions extending the finite bounded theory.
22. The p-integral on [1,∞) has a sharp convergence boundary
Consider ∫₁^∞ x^{-p}dx.
For p≠1, an antiderivative is x^{1−p}/(1−p).
The limit at infinity is finite exactly when 1−p<0, that is p>1.
For p=1, ∫₁^R dx/x=log R→∞.
Therefore
∫₁^∞ x^{-p}dx converges iff p>1.
23. Endpoint singularities have the opposite p-boundary
For ∫₀¹ x^{-p}dx, the problematic point is zero.
The integral converges exactly when p<1.
This contrast prevents a common memory error: the p-threshold depends on whether the singular behaviour is at infinity or near zero.
24. Comparison tests transfer convergence
If 0≤f≤g eventually and ∫g converges, then ∫f converges.
If 0≤g≤f eventually and ∫g diverges, then ∫f diverges.
Limit comparison gives a stronger asymptotic tool: if f/g→c with 0<c<∞, then positive improper integrals f and g have the same convergence behaviour.
25. Absolute convergence is stronger than conditional convergence
If ∫|f| converges, then ∫f converges.
The converse can fail.
Oscillation can create cancellation strong enough for ∫f to converge while ∫|f| diverges.
This distinction becomes even more prominent for infinite series in R25.04.
26. Improper integrals over interior singularities must be split
If f is singular at c∈(a,b), then define
∫_a^b f=∫_a^c f+∫_c^b f
only if both one-sided improper integrals converge separately.
Cancellation across the singularity does not rescue ordinary improper convergence.
A Cauchy principal value is a different construction and must be labelled as such.
27. Worked boundary example: ∫_{−1}^1 1/x dx
The symmetric principal value is zero because the logarithmic divergences cancel under symmetric truncation.
But the ordinary improper integral does not exist: ∫_{−1}^0 1/x dx and ∫_0^1 1/x dx each diverge.
This is a clear example of why a new summation convention must not be confused with the original definition.
28. Integration and differentiation do not always invert each other without hypotheses
FTC gives a powerful inverse relationship for continuous functions and their accumulated integrals.
Outside that setting, one must ask which derivative exists, which integral is being used, and whether equality holds pointwise, almost everywhere or in another sense.
R25.07 will extend integration to Lebesgue theory, where far more functions become integrable and “almost everywhere” statements become central.
29. A dependable integration workflow
First identify the integral type: proper Riemann, improper, or a later measure-theoretic integral.
For proper Riemann integrals, check boundedness and a valid integrability theorem.
Use FTC only after the hypotheses connecting integral and antiderivative are secured.
For substitution, control the change of variable and transformed limits. For integration by parts, track the boundary term.
For improper integrals, isolate every infinite endpoint or singularity and test each limit separately.
30. Common misconceptions
The definite integral is defined as an antiderivative difference. In rigorous Riemann theory it is defined by sums; FTC proves the antiderivative formula.
Every discontinuous function is nonintegrable. False.
An improper integral is just a very large finite integral. No. Its existence is a limit statement.
Cancellation across a singularity proves convergence. Not for ordinary improper integrals.
Area and definite integral are identical. The integral is signed accumulation; total geometric area may require absolute value or splitting.
31. Independent practice: twenty questions
- Define a partition of [a,b].
- What is the mesh of a partition?
- Write a tagged Riemann sum.
- State one epsilon-style definition of Riemann integrability.
- Define upper and lower sums.
- State Darboux’s integrability criterion.
- Why are continuous functions on [a,b] Riemann integrable?
- Why are monotone functions on [a,b] integrable?
- Compute ∫₀¹x dx from equal right-endpoint sums.
- Why can ∫f differ from geometric area?
- State FTC Part I.
- State FTC Part II.
- Explain substitution conceptually.
- State integration by parts.
- Why is the Dirichlet function not Riemann integrable?
- Define ∫₁^∞f as an improper integral.
- For what p does ∫₁^∞x^{-p} converge?
- For what p does ∫₀¹x^{-p} converge?
- Why does ∫_{−1}^1 1/x fail as an ordinary improper integral?
- What is the difference between convergence and absolute convergence?
32. Worked solutions and checks
1. A finite ordered set a=x₀<⋯<x_n=b.
2. max_i(x_i−x_{i−1}).
3. Σf(ξ_i)Δx_i with ξ_i chosen inside each subinterval.
4. All tagged sums with sufficiently small mesh must lie within ε of one common value I.
5. L=Σm_iΔx_i and U=ΣM_iΔx_i using subinterval infima and suprema.
6. f is integrable iff for every ε>0 some partition has U−L<ε.
7. Uniform continuity on the compact interval makes oscillation on sufficiently short subintervals uniformly small.
8. Their total upper-lower gap is controlled by mesh times total monotone variation.
9. The sum equals (n+1)/(2n) and tends to 1/2.
10. Negative function values contribute negative signed accumulation.
11. If F(x)=∫_a^x f(t)dt with f continuous, then F'(x)=f(x).
12. If G’=f with f continuous, then ∫_a^b f=G(b)−G(a).
13. It is a change-of-variables theorem derived from the chain rule and FTC, not literal differential cancellation.
14. ∫uv’=[uv]−∫u’v.
15. Every interval contains rationals and irrationals, so every upper sum is 1 and every lower sum is 0 on [0,1].
16. lim_{R→∞}∫₁^R f, when the finite limit exists.
17. p>1.
18. p<1.
19. The one-sided integrals at zero diverge separately; symmetric principal-value cancellation is a different concept.
20. Absolute convergence means ∫|f| converges; it implies ordinary convergence, while the converse can fail.
33. Where this guide hands off
R25.04 moves from continuously accumulated intervals to discrete infinite accumulation: partial sums, absolute versus conditional convergence, comparison methods and power series.
Sources and further study
Use MIT OpenCourseWare 18.100A Real Analysis for a rigorous real-analysis route. For school and A-Math applications, return to BTT’s separate definite-integral and integration-rule guides rather than treating this page as an examination syllabus owner.