   Chapter 8.6, Problem 5E

Chapter
Section
Textbook Problem

Using the Trapezoidal Rule and Simpson's Rule In Exercises 3-14, use the Trapezoidal Rule and Simpson’s Rule to approximate the value of the definite integral for the given value of n. Round your answer to four decimal places and compare the results with the exact value of the definite integral. ∫ 3 4 1 x − 2 d x ,   n = 4

To determine

To calculate: The approximate value of definite integral 341x2dx for given value of n by using Trapezoidal rule and Simpson’s rule and compare it with the exact value of the definite integral.

Explanation

Given: The provided integral is:

341x2dx

And the value n=4.

Formula used:

1. According to trapezoidal rule

abf(x)dxba2n[f(x0)+2f(x1)+....+2f(xn1)+f(xn)]

2. According to Simpson’s rule

abp(x)dx(ba3n)[f(x0)+4f(x1)+2f(x2)+4f(x3)+...2f(xn2)+4f(xn1)+f(xn)]

3. The log rule for integration:

dxx=log|x|+C

Calculation:

Consider the following function

f(x)=1x2.

From the given integral we get,

b=4, a=3 and n=4.

Now, calculate the width as:

Δx=ban=434=14

Use trapezoidal rule to approximate the value of integral,

341x2dx432(4)[f(3)+2f(3+14)+2f(3+14×2)+2f(3+14×3)+f(3+14×4)]=18[f(3)+2f(134)+2f(72)+2f(154)+f(4)]=18[(132)+2(1(13/4)2)+2(1(7/2)2)+2(1(15/4)2)+(142)]=18[1+2(45)+2(23)+2(47)+12]

By solving further, we get,

18[1+2(45)+2(23)+2(47)+12]=18[1+85+43+87+12]=18[1+1

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