# Definition:Definite Integral/Limits of Integration/Upper Limit

## Definition

Let $a, b \in \R$ be real numbers such that $a \le b$.

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

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

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

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

## 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} .