Question

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible.

• the definite integral of a function f on an interval $[a,b]$as a limit of Riemann sums, including the definitions of $\Delta x$, $x_k$, and $x_k^{*}$
  

Short answer

Ans:${\int }_a^{b}f\left(x\right)dx=\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left({x_k}^{*}\right)\Delta x$

Step-by-step solution

Step 1. Given information: 

[a,b]

Step 2.  Defining with a graph and algebraic example :

Generally, the area under the curve is defined in terms of the limit of sums.

Let f is a continuous function defined on an interval $[a,b],\mathrm{n}$as a positive integer. Then, the definite integral of the function from a to b is

${\int }_a^{b}f\left(x\right)dx=\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left({x_k}^{*}\right)\Delta x$

Here, $\Delta x=\frac{b-a}{n},x_k=a+k\Delta x$.

$x_k{}^{*}\to$is the any sample point in the sub interval