   Chapter 8.6, Problem 12E

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. ∫ 0 2 x x 2 + 1 d x ,   n = 4

To determine

To calculate: The approximate value of definite integral 02xx2+1dx for the specified value of n=4 by Trapezoidal Rule and Simpson’s Rule and compare it with the exact value of the definite integral.

Explanation

Given:

The integral is 02xx2+1dx.

Formula used:

Trapezoidal Rule is:

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

Simpson’s Rule is:

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

Calculation:

Suppose f(x)=xx2+1

Since n=4

Therefore, add 204=12 to each term to get values of x0,x2.......xm where x0=0 and x4=2.

Thus, the values of x are:

x1=0+12=12x2=12+12=1

And,

x3=1+12=32x4=32+12=2

Now, by Trapezoidal Rule:

Here b=2, a=0 and n=4

Therefore,

02xx2+1dx202(4)[f(0)+2f(12)+2f(1)+2f(32)+f(2)]=14[(002+1)+2((12)(1/2)2+1)+2(112+1)+2((32)(3/2)2+1)+(222+1)]=14[0+2((12)54)+2(2)+2((32)134)+(25)]=14[0+52+22+3213+25]

By simplifying more,

02xx2+1dx=14[1

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