91影视

The given data isf f is integrable and monotonically increasing on [a,b].

Consider an integrable function f that is monotonically increasing on [a,b]

Let n be a positive integer.

A function is monotonically increasing on interval [a,b] if $f\left(x\right)鈮�f\left(y\right)\mathrm{whenever}x鈮�y$

Consider an n rectangle partition of [a,b] into subintervals of the form $[x_{k-1},x_k]\mathrm{where}1鈮�k鈮�n$

For each k, the function satisfies the following condition due to monotonicity,

$f\left(x_{k-1}\right)鈮�f\left(x\right)鈮�f\left(x_k\right)$

Define $鈭�x=\frac{b-a}{n}and{x}_k=a+k鈭�x$

Then n -rectangle left sum and n-rectangle right sum is determined as follows,

$LEFT\left(n\right)=\underset{k=1}{\overset{n}{鈭�}}f\left(x_{k-1}\right)鈭�xandRIGHT\left(n\right)=\underset{k=1}{\overset{n}{鈭�}}f\left(x_k\right)鈭�x$

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

In each sub-interval $[x_{k-1},x_k];1鈮�k鈮�n$, the kth left sum rectangle has an area of $f\left(x_{k-1}\right)鈭�x$ and kth right sum rectangle has an area of $f\left(x_k\right)鈭�x$

Thus, in each sub-interval the following inequality holds due to monotonically increment of the function.

localid="1649658120728" $f\left(x_{k-1}\right)鈭�x鈮�f\left(x\right)鈭�x鈮�f\left(x_k\right)鈭�x$

Sum up all the areas in all the n sub-intervals to obtain the following inequality,

localid="1649658152012" $$\begin{gathered} \underset{k=1}{\overset{n}{鈭�}}f\left(x_{k-1}\right)鈭�x鈮�{鈭�}_a^{b}f\left(x\right)dx鈮�\underset{k=1}{\overset{n}{鈭�}}f\left(x_k\right)鈭�x \\ LEFT\left(n\right)鈮�{鈭�}_a^{b}f\left(x\right)dx鈮�RIGHT\left(n\right) \end{gathered}$$

Hence, proved.