   Chapter 5.6, Problem 2CP ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Checkpoint 2 Worked-out solution available at LarsonAppliedCalculus.comUse the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of f ( x )   = 1 / ( x 2 +   2 ) , the x-axis, and the lines x = 0 and x = 1.

To determine

To calculate: The area bounded by the function f(x)=1x2+2 and the line x=0 and x=1 in by mid-point rule. Also plot the graph between function f and x-axis.

Explanation

Given Information:

The provided function is f(x)=1x2+2. area of this function is calculated in the line x=0 and x=1 which means area is calculated in the interval [0,1] and the provided interval will be divided into 4 sub interval as n=4.

Formula used:

The approximate area of any definite integral baf(x)dx by the use of midpoint rule is calculated by the use of following steps.

Step1: firstly, divide the provided interval of the function into n sub intervals by the use of formula;

Δx=ban where [a,b] are the intervals and n is the subinterval value. and Δx is the width.

Step2: by the use of above calculated value of subinterval find the midpoint of each sub interval.

Step3: final step is to obtain approximate area by calculating function f at each mid-point from the use of formula;

abf(x)dx=ban[f(x1)+f(x2)+f(x3)++f(xn)]

Calculation:

Consider the provided function;

Now, put n equal to 4, a=1 and b=1 in the above mid-point rule as;

Δx=104Δx=14

Width Δx of the provided interval will be 14.

So, the interval [0,1] with width 14 will be divided into 4 subintervals as;

[0,14], [14,12], [12,34] and  [34,1]

Now, the value of midpoint of each sub interval is calculated as;

[x1,x2]=x1+x22 where x1,x2 are the upper value and lower value of the subintervals

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