## 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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