Definition:Definite Integral

Let $$\left[{a \,. \, . \, b}\right]$$ be a closed interval of the set $$\mathbb{R}$$ of real numbers.

Let $$f: \mathbb{R} \to \mathbb{R}$$ be a real function.

Let $$f \left({x}\right)$$ be bounded on $$\left[{a \,. \, . \, b}\right]$$.

Suppose that $$\exists y \in \mathbb{R}$$ such that:


 * For any lower sum $$L$$ over any of subdivision of $$\left[{a \, . \, . \, b}\right]$$, $$L \le y$$;
 * For any upper sum $$U$$ over any of subdivision of $$\left[{a \, . \, . \, b}\right]$$, $$U \ge y$$.

Then $$y$$ is known as the ''definite integral of $$f \left({x}\right)$$ over $$\left[{a \,. \, . \, b}\right]$$'' and is denoted $$y = \int_a^b f \left({x}\right) dx$$.

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

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