User:Keith.U/Definite Integral

Definition: Definite Integral
Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval.

Let $f: \left[{a \,.\,.\, b}\right] \to \R$ be a real function.

Riemann Integrable
More usually (and informally), we say:
 * $f$ is (Riemann) integrable over $\left[{a \,.\,.\, b}\right]$.

Riemann Integral
The real number $L$ as defined above is called the Riemann integral of $f$ over $\left[{a \,.\,.\, b}\right]$ and is denoted:
 * $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x$

Darboux Integral
The Darboux integral of $f$ over $\left[{a \,.\,.\, b}\right]$ is denoted
 * $\displaystyle \int_a^b f \left({x}\right) \, \mathrm d x$

and is defined as:
 * $\displaystyle \int_a^b f \left({x}\right) \, \mathrm d x = \underline{\int_a^b} f \left({x}\right) \ \mathrm d x = \overline{\int_a^b} f \left({x}\right) \, \mathrm d x$

Also known as
Many sources whose target consists of students at a relatively elementary level refer to this merely as a definite integral.

Expositions which delve deeper into the structure of integral calculus often establish the concepts of the Riemann integral and the Darboux integral, and contrast them with the Lebesgue integral, which is an extension of the concept into the more general field of measure theory.

Also see

 * Definition:Riemann Sum
 * Definition:Signed Area
 * Continuous Real Function is Darboux Integrable


 * Equivalence of Definitions of Riemann Integral

There are more general definitions of integration; see:
 * Definition:Lebesgue Integral
 * Lebesgue Integral is Extension of Riemann Integral