91Ó°ÊÓ

Let $n$be a positive integer.

A function is monotonically increasing on an interval $[a,b]$if $f\left(x\right)≤f\left(y\right)$whenever $x≤y$; where $x$and $y$are elements in $[a,b]$.

Consider an $n$-rectangle partition of $[a,b]$into subintervals of the form $\left[x_{k-1},x_k\right]$, where $1≤k≤n$.

Consider the integrable function $f$that is monotonically increasing on an interval $[a,b]$.

For each $k$, the function $f$satisfies the following condition due to monotonicity: $f\left(x_{k-1}\right)≤f\left(x\right)≤f\left(x_k\right)$, for all $x$in $\left[x_{k-1},x_k\right].$

Define $\mathrm{Δ}x=\frac{b-a}{n}$and $x_k=a+k\mathrm{Δ}x$.

The $n$-rectangle left-sum is determined as $LEFT\left(n\right)=\underset{k=1}{\overset{n}{∑}}f\left(x_{k-1}\right)\mathrm{Δ}x$.

The $n$-rectangle right-sum is determined as $RIGHT\left(n\right)=\underset{k=1}{\overset{n}{∑}}f\left(x_k\right)\mathrm{Δ}x$.

Now, the area under the curve of the function $f$is determined by the definite integral ${∫}_a^{b}f\left(x\right)dx$

In each subinterval $\left[x_{k-1},x_k\right];1≤k≤n$, the $k$th left-sum rectangle has an area of $f\left(x_{k-1}\right)\mathrm{Δ}x$, and the $k$th right-sum rectangle has an area of $f\left(x_k\right)\mathrm{Δ}x$.

Therefore, in each subinterval $\left[x_{k-1},x_k\right]$, the following inequality holds due to monotonicity of the function $f$:

$f\left(x_{k-1}\right)\mathrm{Δ}x≤{∫}_{x_{k-1}}^{x_1}f\left(x\right)dx≤f\left(x_k\right)\mathrm{Δ}x$

Here, ${∫}_{x_{k-1}}^{x_1}f\left(x\right)dx$is the area under the curve of the function $f$in the subinterval $\left[x_{k-1},x_k\right]$.

Sum up the areas in all the $n$subintervals to obtain the following inequality:

$\underset{k=1}{\overset{n}{∑}}f\left(x_{k-1}\right)\mathrm{Δ}x≤{∫}_a^{b}f\left(x\right)dx≤\underset{k=1}{\overset{n}{∑}}f\left(x_k\right)\mathrm{Δ}x$

That means, $LEFT\left(n\right)≤{∫}^{b}f\left(x\right)dx≤RIGHT\left(n\right)$.

Hence, it is proved that for any positive integer $n,LEFT\left(n\right)≤{∫}_a^{b}f\left(x\right)dx≤RIGHT\left(n\right)$