Definition:Definite Integral/Limits of Integration

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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 $\left[{a \,.\,.\, b}\right]$.

Let the definite integral of $f$ with respect to $x$ from $a$ to $b$ be:

$\displaystyle \int_a^b f \left({x}\right) \, \mathrm d x$


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 $b$


From the Fundamental Theorem of Calculus, we have that:

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

where $F$ is a primitive of $f$, that is:

$f \left({x}\right) = \dfrac {\mathrm d} {\mathrm d x} F \left({x}\right)$


Then $F \left({b}\right) - F \left({a}\right)$ is usually written:

$\Big[{ F \left({x}\right) }\Big]_a^b := F \left({b}\right) - F \left({a}\right)$

or:

$\Big.{ F \left({x}\right) }\Big|_a^b := F \left({b}\right) - F \left({a}\right)$


Also known as

The interval defined by the limits of integration can be referred to as the range of integration.


Sources