Truncating at the third term yields the expression $int_0^1 (1-u^n)^M ,du = frac{Gamma(1+1/n)}{M^{1/n}}left{1 – frac{n+1}{2n^2M} + Oleft(frac{1}{M^2}right)right} hspace{1cm} (M to infty)$, where the use of side-set or limits of integration below/above is not explicitly defined by any design rules. As noted by @PeterGrill and the other answer, setting the limit of integration below the integral symbol(s) may be appropriate in cases of multiple integrals or when expressing the entire set over which the integration takes place with a symbol (e.g., ).

Solution 1:

After conducting research online, it appears that utilizing both

```
``` overline

as well as

```
``` underline

is adequate.

Below is a concise illustration that outlines the definition of.

`upRiemannint{`}{}

The “upper Riemann integral” is calculated for the interval between

<lo>

and

<hi>

. This process is analogous to the previous step.

`loRiemannint{`}{}

Specifies the integral known as the “lower Riemann integral” across the interval starting from

<lo>

and ending at

<hi>

.

```
documentclass{article}
newcommand{upRiemannint}[2]{
overline{int_{#1}^{#2}}
}
newcommand{loRiemannint}[2]{
underline{int_{#1}^{#2}}
}
begin{document}
[
loRiemannint{a}{b} f(x),mathrm{d}x qquad textrm{or} qquad upRiemannint{a}{b} f(x),mathrm{d}x
]
end{document}
```

The use of these integrals in text mode is also possible, but the default integral sign causes a slight misalignment in the vertical position.

Solution 2:

In my opinion, opting for shorter bars leads to increased complexity in the macro.

```
documentclass{article}
usepackage{amsmath}
defupint{mathchoice%
{mkern13muoverline{vphantom{intop}mkern7mu}mkern-20mu}%
{mkern7muoverline{vphantom{intop}mkern7mu}mkern-14mu}%
{mkern7muoverline{vphantom{intop}mkern7mu}mkern-14mu}%
{mkern7muoverline{vphantom{intop}mkern7mu}mkern-14mu}%
int}
deflowint{mkern3muunderline{vphantom{intop}mkern7mu}mkern-10muint}
begin{document}
begin{gather*}
upint_a^b f(x),mathrm{d}x \
lowint_a^b f(x),mathrm{d}x
end{gather*}
end{document}
```

If you wish to have full authority on the positioning of the bars, you can experiment with the subsequent code for the lower integral.

```
lefteqn{int_a^b f(x)}lefteqn{hspace{0.0ex}rule[-2.25ex]{1.1ex}{.05ex}}
phantom{int_a^b f(x)}mathrm{d}x
```

And for the upper integral:

```
lefteqn{int_a^b f(x)}lefteqn{hspace{1.2ex}rule[ 3.35ex]{1.1ex}{.05ex}}
phantom{int_a^b f(x)}mathrm{d}x
```

The positioning of the bar can be controlled using the

hspace

argument while the height, length, and thickness can be controlled using the three

rule

arguments.