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Subsection: Riemann Integral

This subsection of BookOfProofs is dedicated to the Riemann integral, which is defined on some closed real intervals $[a,b]$. The Riemann integral $\int_a^b f(x)dx$ can be interpreted as the area enclosed by the $x$-axis and the graph of the function $f$ on the interval $[a,b]$.

In some sense, the integration is inverse to the differentiation, which is shown in the corresponding theorem. This fact allows in many cases to calculate the integral of a function using an explicit formula.

Table of Contents

  1. Proposition: Riemann Integral for Step Functions
  2. Definition: Riemann-Integrable Functions
  3. Proposition: Riemann Upper and Riemann Lower Integrals for Bounded Real Functions
  4. Definition: Riemann Sum With Respect to a Partition
  5. Theorem: Indefinite Integral, Antiderivative
  6. Theorem: Fundamental Theorem of Calculus
  7. Proposition: Integrals on Adjacent Intervals
  8. Theorem: Integration by Substitution
  9. Theorem: Mean Value Theorem For Riemann Integrals
  10. Theorem: Partial Integration
  11. Lemma: Riemann Integral of a Product of Continuously Differentiable Functions with Sine
  12. Lemma: Trapezoid Rule

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