Definition:Definite Integral

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

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

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

Suppose that $\exists y \in \R$ 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:
 * $\displaystyle y = \int_a^b f \left({x}\right) \ \mathrm d x$

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

$f \left({x}\right)$ 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:


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

Geometric Interpretation
The expression $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x$ can be (and frequently is) interpreted as the area under the graph. This follows from the definition of the definite integral as a sum of the product of the lengths of intervals and the "height" of the function being integrated in that interval and the formula for the area of a rectangle.

A depiction of the lower and upper sums illustrates this:


 * RiemannLowerSum.png RiemannUpperSum.png

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 signed area under the graph becomes negative.

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

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

Historical Note
Consider the Riemann sum:


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

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 $\mathrm dx$


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

Hence the similarity in notation:


 * $\displaystyle \sum_a^b \ f\left({x}\right) \ \Delta x \to \int_a^b f\left({x}\right) \ \mathrm dx$

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 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.