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Riemann Integral

This subsection of bookofproofs.org 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.

| | | | Contributors: bookofproofs

1.Definition: Riemann Integrable Functions

2.Definition: Riemann Sum With Respect to a Partition

3.Theorem: Indefinite Integral, Antiderivative

4.Theorem: Mean Value Theorem For Riemann Integrals

5.Theorem: Fundamental Theorem of Calculus

6.Theorem: Integration by Substitution

7.Proposition: Riemann Integral for Step Functions

8.Proposition: Riemann Upper and Riemann Lower Integrals for Bounded Real Functions

9.Proposition: Integrals on Adjacent Intervals


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