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

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

It follows from Upper Sum Never Smaller than Lower Sum that $$\sup L \left({P}\right) = \int_a^b f \left({x}\right) dx = \inf U \left({P}\right)$$.

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

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

If $$a > b$$ then we define $$\int_a^b f \left({x}\right) dx = - \int_b^a f \left({x}\right) dx$$.

Limits of Integration
In the expression $$\int_a^b f \left({x}\right) dx$$, the values $$a$$ and $$b$$ are called the limits of integration

If there is no danger of confusing the concept with limit of a function or of a sequence, just limits.

Thus $$\int_a^b f \left({x}\right) dx$$ can be voiced: "The integral of (the function) $$f$$ of $$x$$ with respect to $$x$$ (evaluated) between the limits (of integration) $$a$$ and $$b$$".

More compactly (and usually), it is voiced "The integral of $$f$$ of $$x$$ w.r.t. $$x$$ between $$a$$ and $$b$$" or "The integral of $$f$$ of $$x$$ dee $$x$$ from $$a$$ to $$b$$."

Geometric Interpretation
The expression $$\int_a^b f \left({x}\right) dx$$ can be (and frequently is) interpreted as the area under the graph.

A depiction of the lower and upper sums illustrates this:



It can intuitively be seen that as the number of points in the subdivision increases, the more "accurate" the lower and upper sums become.

Also note that if the graph is below the $$x$$-axis, the area under the graph becomes negative.