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.

- Proposition: Riemann Integral for Step Functions
- Definition: Riemann-Integrable Functions
- Proposition: Riemann Upper and Riemann Lower Integrals for Bounded Real Functions
- Definition: Riemann Sum With Respect to a Partition
- Theorem: Indefinite Integral, Antiderivative
- Theorem: Fundamental Theorem of Calculus
- Proposition: Integrals on Adjacent Intervals
- Theorem: Integration by Substitution
- Theorem: Mean Value Theorem For Riemann Integrals
- Theorem: Partial Integration
- Lemma: Riemann Integral of a Product of Continuously Differentiable Functions with Sine
- Lemma: Trapezoid Rule