Chapter 13.5, Problem 1E

The graph of a function f is shown below.

1. To find the area under the graph of f, we first approximate the area by. Approximate the area by drawing four rectangles. The area $R_4$of this approximation is

$R_4=\boxed{}+\boxed{}+\boxed{}+\boxed{}$

To determine

The first step to calculate the area under the graph of f and find the area $R_4$ for the given graph.

Answer to Problem 1E

The first step to calculate the area under the graph of f is approximate the area by using rectangles and the area $R_4$ for the given graph is $f\left(x_1\right)\left(x_1-a\right)+f\left(x_2\right)\left(x_2-x_1\right)+f\left(x_3\right)\left(x_3-x_2\right)+f\left(b\right)\left(b-x_3\right)$.

Explanation of Solution

The given figure shows the graph of the function f,

Figure (1)

First divide the area below the graph of the function f by n strips or rectangles of equal width, to find the area under the graph of f.

The width of the each n rectangle in the interval $\left[a,b\right]$ is given below,

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

The area of the $k\text{th}$ rectangle is calculated as given below,

$\text{Area}\text{of}k\text{th}\text{rectangle}=f\left(x_k\right)\Delta x$.

The total area under the graph of the function f is calculated as,

$\text{Area}=f\left(x_1\right)\Delta x+f\left(x_2\right)\Delta x+\cdots +f\left(x_n\right)\Delta x$ (1)

From figure (1), it is noticed that the graph of f lies between the interval $\left[a,b\right]$. Divide the graph in four rectangles with help of three points $x_1,x_2$ and $x_3$.

The width of the rectangles are $\left(x_1-a\right),\left(x_2-x_1\right),\left(x_3-x_2\right)$ and $\left(b-x_3\right)$ respectively.

Use the equation (1) and the width of the rectangles to find the area of the under the graph of f.

$\text{Area}=f\left(x_1\right)\left(x_1-a\right)+f\left(x_2\right)\left(x_2-x_1\right)+f\left(x_3\right)\left(x_3-x_2\right)+f\left(b\right)\left(b-x_3\right)$

Therefore, the first step to calculate the area under the graph of f is approximate the area by using rectangles and the area $R_4$ for the given graph is $f\left(x_1\right)\left(x_1-a\right)+f\left(x_2\right)\left(x_2-x_1\right)+f\left(x_3\right)\left(x_3-x_2\right)+f\left(b\right)\left(b-x_3\right)$.