Definition:Definite Integral

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

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

Suppose that:
 * $\displaystyle \underline{\int_a^b} f \left({x}\right) \ \mathrm dx = \overline{\int_a^b} f \left({x}\right) \ \mathrm dx$

where $\displaystyle \underline{\int_a^b}$ and $\displaystyle \overline{\int_a^b}$ denote the lower integral and upper integral, respectively.

Then the definite (Riemann) integral of $f$ over $\left[{a \,.\,.\, b}\right]$ is defined as:
 * $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x = \underline{\int_a^b} f \left({x}\right) \ \mathrm dx = \overline{\int_a^b} f \left({x}\right) \ \mathrm dx$

$f$ is formally defined as (properly) integrable over $\left[{a \,.\,.\, b}\right]$ in the sense of Riemann, or (properly) Riemann integrable over $\left[{a \,.\,.\, b}\right]$.

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

If $a > b$, then we define:


 * $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x = - \int_b^a f \left({x}\right) \ \mathrm d x$

Integrand
In the expression $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x$, the function $f$ is called the integrand.

This term comes from the cod-Latin for that which is to be integrated.

Historical Note
Consider the Riemann sum:


 * $\displaystyle \sum_{i \mathop = 1}^n \ f\left({c_i}\right) \ \Delta x_i$

Historically, the definite integral was an extension of this type of sum such that:


 * The finite distance $\Delta x$ is instead the infinitely small distance $\mathrm d x$


 * The finite sum $\Sigma$ is instead the sum of an infinite amount of infinitely small quantities: $\int$

Hence the similarity in notation:



The notion of "infinitely small" does not exist in the modern formulation of real numbers. Nevertheless, this idea is sometimes used as an informal interpretation of the definite integral.

Also known as
This definition of the Riemann integral is also known as the Darboux integral.

Also see

 * Riemann sum
 * Signed area

Note that a continuous function is always Riemann integrable.

There are more general definitions of integration; see Lebesgue Integral is Extension of Riemann Integral.