Definition:Definite Integral/Limits of Integration

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Definition

Let $\closedint a b$ be a closed interval of the set $\R$ of real numbers.

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

Let $\map f x$ be bounded on $\closedint a b$.

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

$\ds \int_a^b \map f x \rd x$


In the expression $\ds \int_a^b \map f x \rd x$, 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 $\ds \int_a^b \map f x \rd x$ 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 $b$


Lower Limit

The limit $a$ is referred to as the lower limit of the integral.


Upper Limit

The limit $b$ is referred to as the upper limit of the integral.


Also known as

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

Some sources refer to it as the interval of integration.


Also see

From the Fundamental Theorem of Calculus, we have that:

$\ds \int_a^b \map f x \rd x = \map F b - \map F a$

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

$\map f x = \dfrac \d {\d x} \map F x$


Then $\map F b - \map F a$ is usually written:

$\bigintlimits {\map F x} {x \mathop = a} {x \mathop = b} := \map F b - \map F a$

or, when there is no chance of ambiguity as to the independent variable:

$\bigintlimits {\map F x} a b := \map F b - \map F a$


Some sources use:

$\bigvalueat {\map F x} a^b := \map F b - \map F a$

but this is not recommended, as it is not so clear exactly where the expression being evaluated actually starts.


Technical Note

The $\LaTeX$ code for \(\intlimits {\dfrac {\map f s} s} {s \mathop = 1} {s \mathop = a}\) is \intlimits {\dfrac {\map f s} s} {s \mathop = 1} {s \mathop = a} .


When the expression being evaluated fits into the line and does not expand upwards or downwards much, the square brackets become similarly small, so making the expression difficult to read, thus:

The $\LaTeX$ code for \(\intlimits {\map f s} {s \mathop = 1} {s \mathop = a}\) is \intlimits {\map f s} {s \mathop = 1} {s \mathop = a} .


Hence we have another command that uses bigger square brackets:

The $\LaTeX$ code for \(\bigintlimits {\map f s} {s \mathop = 1} {s \mathop = a}\) is \bigintlimits {\map f s} {s \mathop = 1} {s \mathop = a} .


Sources

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