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Chapter 8: Integration

The definition of integration is quite complicated and we devote some time to studying that. We identify a class of "integrable" functions wihich is analogous to the "differentiable" functions of Chapter 4, and show that all continuous functions are integrable. Evaluating the integral based on the definition alone is very hard, so we prove the Fundamental theorem of calculus; the theoretical tool which is the basis of all the integration techniques in calculus. We then develop a more general theory of integration called Riemann-Stieltjes integration. This enables us to unify integration and summation; something which is useful in applications in mechanics and statistics, to name two.

Chapter Contents

• Class 50
  • The Definition of the Riemann Integral
• Class 51
  • Properties of the Integral
• Class 52
  • Evaluating the Integral
• Class 53
  • More Properties of the Integral
  • The Indefinite Integral, and Another Form of the Fundamental Theorem
  • Riemann Integrable Functions
• Class 54
  • Riemann-Stieltjes Integration
• Class 55