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

Limits of Integration
In the expression $\displaystyle \int_a^b f \left({x}\right) \mathrm d x$, 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 $\displaystyle \int_a^b f \left({x}\right) \mathrm d x$ 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 $x$ with respect to $x$ between $a$ and $b$

or:
 * The integral of $f$ of $x$ dee $x$ from $a$ to $b$

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 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.