   Chapter 8.6, Problem 9E

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. ∫ 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:

49xdx

And,  n=8

Formula used:

As per trapezoidal rule:

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

As per Simpson’s rule:

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

Calculation:

Suppose f(x)=x

As n=8

Hence, add 948=58 to each term to find values of x0,x1.......xm where x0=4 and 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:

49xdx942(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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