# 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 $\closedint a b$, $\Delta = \set {x_0, \ldots, x_n}$, $x_0 = a$ and $x_n = b$.

Let there for $\Delta$ be a corresponding sequence $C$ of sample points $c_i$, $C = \tuple {c_1, \ldots, c_n}$, where $c_i \in \closedint {x_{i - 1} } {x_i}$ for every $i \in \set {1, \ldots, n}$.

Let $\map S {f; \Delta, C}$ 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 $\closedint a b$ if and only if:

- $\exists L \in \R: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall$ finite subdivisions $\Delta$ of $\closedint a b: \forall$ sample point sequences $C$ of $\Delta: \norm \Delta < \delta \implies \size {\map S {f; \Delta, C} - L} < \epsilon$

where $\norm \Delta$ denotes the norm of $\Delta$.

The real number $L$ is called the **Riemann integral** of $f$ over $\closedint a b$ and is denoted:

- $\displaystyle \int_a^b \map f x \rd x$

## Subcategories

This category has the following 21 subcategories, out of 21 total.

### D

### L

### M

## Pages in category "Definite Integrals"

The following 47 pages are in this category, out of 47 total.

### C

### D

- Definite Integral from 0 to 1 of Power of u over 1 + Power of u
- Definite Integral from 0 to 1 of x to the minus x
- Definite Integral from 0 to 1 of x to the x
- Definite Integral from 0 to 1 of x^4 (1 - x)^4 over (1 + x^2)
- 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 Fourier Series at Ends of Interval
- Definite Integral of Function satisfying Dirichlet Conditions is Continuous
- Definite Integral of Odd Function
- Definite Integral of Partial Derivative
- 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

### I

### L

### S

- Simpson's Rule
- Sophomore's Dream
- Sum of Integrals on Adjacent Intervals for Integrable Functions
- Sum of Integrals on Adjacent Intervals for Integrable Functions/Corollary
- Sum of Reciprocals of Powers of Odd Integers Alternating in Sign
- Sum of Reciprocals of Powers of Odd Integers Alternating in Sign/Corollary