Q. 2 - 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.

• the geometric interpretations of the $n$-rectangle left sum, right sum, and midpoint sum for a function $f$ on an interval $[a, b]$
  

Step-by-Step Solution

Answer

Ans: 

Part (a). The right Riemann sum: ${\int }_a^{b}f\left(x\right)dx=\sum _{k=1}^{n}f\left(x_k\right)\Delta x$

Part (b). The left Riemann sum:  ${\int }_a^{b}f\left(x\right)dx=\sum _{k=1}^{n}f\left(x_{k-1}\right)\Delta x$

Part (c). The midpoint Riemann sum: ${\int }_a^{b}f\left(x\right)dx=\sum _{k=1}^{n}f\left(x_k^{*}\right)\Delta x$

Step 1. Given information:

The geometrical interpretation of$n$ the rectangle left, right, and midpoint sums the function over an interval $[a, b]$.

Step 2. Finding the right Riemann sum:

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

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

n number of subintervals with equal width $\Delta x$ .

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

$x_k$ is the $k^{th}$  is any sample point in the subinterval

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

The right Riemann sum is

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

If the rectangles are touching the curve with their top right corner, then the area under the curve is called the right sum.

Step 3. Finding the left Riemann sum:

The left Riemann sum is

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

If the rectangles are touching the curve with their top left corner, then the area under the curve is called the left sum.

Step 4. Finding the midpoint Riemann sum

The midpoint Riemann sum is

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

Here, $x_k{}^{*}=\frac{x_i+x_{i+1}}{2}$

If the rectangles are touching the curve with their top midpoint, then the area under the curve is called the midpoint sum.