Chapter 15.3, Problem 2CFU

To determine

Explain the step by step process to find the area under the given graph and between the x-axis values.

Explanation of Solution

Given:

The given function is $y=f\left(x\right)$ between $x=a$ and $x=b$ .

Calculation:

Graph of the function is

  

In the graph, $f\left(x_1\right),f\left(x_2\right),f\left(x_3\right)---f\left(x_n\right)$ are the height of the first rectangle, second rectangle, third rectangle $---$ nth rectangle respectively.

The length of the entire interval from $a$ to $b$ $=b-a$

The width of each of the $n$ rectangles $=\frac{b-a}{n}$ , it is denoted by $\Delta x$ .

  $\Delta x=\frac{b-a}{n}$

The area of the first rectangle $=f\left(x_1\right)\Delta x$

The area of the second rectangle $=f\left(x_2\right)\Delta x$

  $\begin{array}{l}···\\ ···\\ ···\end{array}$

The area of the $n\text{th}$ rectangle $=f\left(x_n\right)\Delta x$

Total area $A_n=f\left(x_1\right)\Delta x+f\left(x_2\right)\Delta x+f\left(x_3\right)\Delta x+---+f\left(x_n\right)\Delta x={\displaystyle \sum _{i=1}^{n}f\left(x_i\right)\Delta x}$

To make the width of the rectangles approach $0$ , let the number of rectangles approach infinity.

Therefore, the exact area of the region under the graph of the function is

  $\underset{n\to \infty }{\lim }{A}_n=\underset{n\to \infty }{\lim }{\displaystyle \sum _{i=1}^{n}f\left(x_i\right)\Delta x}$

This limit is called a definite integral and is denoted by

  ${\displaystyle {\int }_a^{b}f\left(x\right)}dx$

So, the exact area formula for the region under the graph of the function is

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

Hence the steps are given below.

First find $\Delta x=\frac{b-a}{n}$ .

Then find $x_i$ .

  $x_i=a+i\Delta x$

In third step, substitute the value of $\Delta x$ and $x_i$ in the formula then find the exact area.