# Category:Definite Integrals

This category contains results about Definite Integrals.

Definitions specific to this category can be found in Definitions/Definite Integrals.

Let $\Delta$ be a finite subdivision of $\left[{a \,.\,.\, b}\right]$, $\Delta = \left\{{x_0, \ldots, x_n}\right\}$, $x_0 = a$ and $x_n = b$.

Let there for $\Delta$ be a corresponding sequence $C$ of sample points $c_i$, $C = \left({c_1, \ldots, c_n}\right)$, where $c_i \in \left[{x_{i - 1} \,.\,.\, x_i}\right]$ for every $i \in \left\{{1, \ldots, n}\right\}$.

Let $S \left({f; \Delta, C}\right)$ denote the Riemann sum of $f$ for the subdivision $\Delta$ and the sample point sequence $C$.

Then $f$ is said to be **(properly) Riemann integrable** on $\left[{a \,.\,.\, b}\right]$ if and only if:

- $\exists L \in \R: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall$ finite subdivision $\Delta$ of $\left[{a \,.\,.\, b}\right]: \forall$ sample point sequences $C$ of $\Delta: \left\Vert{\Delta}\right\Vert < \delta \implies \left\vert{S \left({f; \Delta, C}\right) - L}\right\vert < \epsilon$

where $\left\Vert{\Delta}\right\Vert$ denotes the norm of $\Delta$.

The real number $L$ is called the **Riemann integral** of $f$ over $\left[{a \,.\,.\, b}\right]$ and is denoted:

- $\displaystyle \int_a^b f \left({x}\right) \rd x$

## Subcategories

This category has the following 10 subcategories, out of 10 total.

### D

### L

## Pages in category "Definite Integrals"

The following 39 pages are in this category, out of 39 total.

### D

- Definite Integral from 0 to a of Reciprocal of Root of a Squared minus x Squared
- Definite Integral from 0 to a of Root of a Squared minus x Squared
- Definite Integral from 0 to a of x^m by (a^n - x^n)^p
- Definite Integral from 0 to Infinity of x^m over (x^n + a^n)^r
- Definite Integral of Constant
- Definite Integral of Constant Multiple of Real Function
- Definite Integral of Even Function
- Definite Integral of Odd Function
- Definite Integral of Partial Derivative
- Definite Integral of Power of u over 1 + Power of u over 0 to 1
- Definite Integral of Reciprocal of Root of a Squared minus x Squared
- Definite Integral of Step Function
- Definite Integral on Zero Interval
- Definite Integral to Infinity of Power of x over 1 + 2 x Cosine Beta + x Squared
- Definite Integral to Infinity of Power of x over 1 + x
- Definite Integral to Infinity of Power of x over Power of x plus Power of a
- Definite Integral to Infinity of Reciprocal of 1 plus Power of x