Q. 1 - Calculus

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.

• a Riemann sum for a function $f$ on an interval $[a, b]$, including the definitions of $\Delta x,x_k$, and $x_k^{*}$
  

Step-by-Step Solution

Answer

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

Step 1. Given information:

The Riemann sum for a function $f$ on an interval $[a, b]$, including the definitions of $\Delta x,x_k$, and $x_k^{*}$

Step 2. Defining with a graph or algebraic example:

Generally, the Riemann sum approximates the area under the curve.

Consider the function $\mathrm{f}$ is defined over an interval $[a, b]$. The interval is divided into

n number of subintervals 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

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

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