Chapter 8.6, Problem 9E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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. ∫ 4 9 x d x ,   n = 8

To determine

To calculate: The approximate value of definite integral 49xdx for the specified value of n by applying Trapezoidal Rule and Simpson’s Rule and evaluate it with the exact value of the definite integral.

Explanation

Given:

The stated integral is:

âˆ«49xdx

And, Â n=8

Formula used:

As per trapezoidal rule:

âˆ«abf(x)dxâ‰ˆbâˆ’a2n[f(x0)+2f(x1)+....+2f(xnâˆ’1)+f(xn)]

As per Simpsonâ€™s rule:

âˆ«abp(x)dxâ‰ˆ(bâˆ’a3n)[f(x0)+4f(x1)+2f(x2)+4f(x3)+...2f(xnâˆ’2)+4f(xnâˆ’1)+f(xn)]

Calculation:

Suppose f(x)=x

As n=8

Hence, add 9âˆ’48=58 to each term to find values of x0,x1.......xm where x0=4Â andÂ x8=9

Therefore, the values of x are:

x1=4+58=378x2=378+58=214

x3=214+58=478x4=478+58=264

Then,

x5=264+58=578x5=578+58=314

Now,

x7=314+58=678x8=678+58=9

Hence, by trapezoidal rule:

Here b=9, a=4 and n=8

Thus, the integral becomes:

âˆ«49xdxâ‰ˆ9âˆ’42(8)[f(4)+2f(378)+2f(214)+2f(478)+2f(264)+2f(578)+2f(314)+2f(678)+f(9)]=516[(4)1/2+2(378)1/2+2(214)1/2+2(478)1/2+2(264)1/2+2(578)1/2+2(314)1/2+2(678)1/2+(9)1/2]=516[2+4

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