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Home/Calculus/Chapter 6: Definite Integration
Chapter 6
17-22 hours total
5 Sections

Definite Integration

The definite integral gives precise meaning to “area under a curve” through Riemann sums, and reveals the profound connection between differentiation and integration via the Newton-Leibniz Formula.

Riemann Sums

Build integrals from limits of sums

Newton-Leibniz

Connect derivatives and integrals

Wallis Formula

Classic integral techniques

MVT for Integrals

Bonnet and Weierstrass

Prerequisites
Chapter 5: Indefinite Integration

Antiderivatives, integration techniques

Chapter 4: Differentiation

Derivatives, mean value theorems

Chapter 3: Function Limits

Continuity, uniform continuity

Chapter 2: Sequences

Limits, supremum/infimum

Chapter Sections

1
CALC-6.1
4-5 hours

The Riemann Integral

Riemann sums, Darboux upper/lower sums, definition of definite integral, and integrability conditions.

Partitions and Riemann sumsDarboux upper and lower sumsDefinition of definite integralIntegrability criteriaClasses of integrable functions
2
CALC-6.2
3-4 hours

Properties of Definite Integrals

Linearity, interval additivity, comparison theorems, and absolute value integrability.

Linearity propertiesOrder preservationInterval additivityAbsolute value integrabilityProduct integrability
3
CALC-6.3
4-5 hours

Fundamental Theorem of Calculus

The profound Newton-Leibniz formula connecting differentiation and integration.

Variable upper limit integralsContinuity of integral functionsDerivative of integral functionsNewton-Leibniz formulaApplications
4
CALC-6.4
3-4 hours

Computation Techniques

Substitution, integration by parts, symmetry techniques, and Wallis/Stirling formulas.

Substitution in definite integralsIntegration by partsSymmetry techniquesWallis formulaStirling formula
5
CALC-6.5
3-4 hours

Mean Value Theorems for Integrals

First and second mean value theorems: Bonnet and Weierstrass forms.

First mean value theoremSecond mean value theorem (Bonnet)Weierstrass formApplications to limitsAbel transformation
Key Formulas Preview

Riemann Sum

S_Δ = Σf(ξₖ)Δxₖ

Darboux Sums

S̄ = ΣMₖΔxₖ, S = Σmₖ Δxₖ

Newton-Leibniz Formula

∫ₐᵇ f(x)dx = F(b) - F(a)

FTC Part 1

d/dx ∫ₐˣ f(t)dt = f(x)

First Mean Value Theorem

∫ₐᵇ f(x)g(x)dx = f(ξ)∫ₐᵇ g(x)dx

Wallis Formula

Iₙ = (n-1)/n · Iₙ₋₂

Learning Path
1

Master Riemann/Darboux sums

Understand how definite integrals are constructed as limits

2

Learn integral properties

Linearity, additivity, and comparison theorems

3

Prove Newton-Leibniz formula

The bridge between differentiation and integration

4

Practice computation techniques

Substitution, parts, Wallis and Stirling formulas

5

Apply mean value theorems

Bonnet and Weierstrass forms for advanced proofs