Chapter B, Problem 1E

Using Rectangles to Approximate the Area of a Region In Exercises 1 and 2, use the rectangles to approximate the area of the region. See Example 1.

$y=x+1$

To determine

To calculate: The approximate area of the region for the equation $y=x+1$ from the provided graph shown below,

Answer to Problem 1E

Solution:

The approximate area of the provided region is 17.5 square units.

Explanation of Solution

Given Information:

A region lying between the graph of $y=x+1$ and the x-axis, between $x=0$ and $x=5$ as shown in graph provided below,

Formula used:

The area of a rectangle with height h and width w is,

$A=w\times h$

Calculation:

Consider the provided region of the graph,

Use five rectangles in the provided figure to approximate the area of the provided region.

Now, compute the heights of the rectangles by evaluating the function $f\left(x\right)=x+1$ at each of the mid-points of the subintervals $\left[0,1\right],\left[1,2\right],\left[2,3\right],\left[3,4\right]\text{ and }\left[4,5\right]$.

As the width of each rectangle is 1, the sum of the areas of the five rectangles is,

$\begin{array}{c}S=w_1{h}_1+w_2{h}_2+w_3{h}_3+w_4{h}_4+w_5{h}_5\\ =1\times f\left(\frac{1}{2}\right)+1\times f\left(\frac{3}{2}\right)+1\times f\left(\frac{5}{2}\right)+1\times f\left(\frac{7}{2}\right)+1\times f\left(\frac{9}{2}\right)\\ =1\times \left(\frac{1}{2}+1\right)+1\times \left(\frac{3}{2}+1\right)+1\times \left(\frac{5}{2}+1\right)+1\times \left(\frac{7}{2}+1\right)+1\times \left(\frac{9}{2}+1\right)\\ =\frac{3}{2}+\frac{5}{2}+\frac{7}{2}+\frac{9}{2}+\frac{11}{2}\end{array}$

Simplify further,

$\begin{array}{c}S=\frac{35}{2}\\ =17.5\text{ square units}\end{array}$

So, the approximate area of the provided region is 17.5 square units.