Chapter 4, Problem 1D

To determine

To give: the mathematical definition of Riemann sum of the function f over an interval

  $[a,b].$

Explanation of Solution

Given information:

Consider the function f over an interval $[a,b].$

Generally, the Riemann sum is the approximating the area under the curve.

  

Consider the function f is defined over an interval $[a,b].$ The interval is divided in to

n number of sub intervals with equal width $\Delta x.$ .

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

Here, $x_k$ is the $k^{th}$ end point of the subintervals. $x_k=a+k\Delta x$

Then the Riemann sum

  ${\displaystyle \underset{a}{\overset{b}{\int }}f(x)dx}={\displaystyle \sum _{k=1}^{n}f(x_k{}^{*})}\Delta x$

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