Trapezoidal Rule Integration

Integration 

Topic: Trapezoidal Rule 

Major: General Engineering 

Author: Autar Kaw, Charlie Barker 

http://numericalmethods.eng.usf.edu 1

Integration: 

The process of measuring 

the area under a function 

plotted on a graph. 

Where: 

f(x) is the integrand 

a= lower limit of integration 

b= upper limit of integration 

2 

What is Integration 

http:// 

numericalmethods.eng.usf.edu

3 

Basis of Trapezoidal Rule 

Trapezoidal Rule is based on the Newton-Cotes 

Formula that states if one can approximate the 

integrand as an n th order polynomial… 

and 

where 

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4 

Basis of Trapezoidal Rule 

Then the integral of that function is approximated 

by the integral of that n th order polynomial. 

Trapezoidal Rule assumes n=1, that is, the area 

under the linear polynomial, 

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Derivation of the Trapezoidal Rule 

5 

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6 

Method Derived From Geometry 

The area under the 

curve is a trapezoid. 

The integral 

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7 

Example 1 

The vertical distance covered by a rocket from t=8 to 

t=30 seconds is given by: 

a) Use single segment Trapezoidal rule to find the distance covered. 

b) Find the true error, for part (a). 

c) Find the absolute relative true error, for part (a). 

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a) 

8 

Solution 

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a) 

9 

Solution (cont) 

b) The exact value of the above integral is 

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) 

10 


c) The absolute relative true error, , would be 

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11 

Multiple Segment Trapezoidal Rule 

In Example 1, the true error using single segment trapezoidal rule was 

large. We can divide the interval [8,30] into [8,19] and [19,30] intervals 

and apply Trapezoidal rule over each segment. 

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Hence: 

12 

With 


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13 


The true error is: 

The true error now is reduced from -807 m to -205 m. 

Extending this procedure to divide the interval into equal 

segments to apply the Trapezoidal rule; the sum of the 

results obtained for each segment is the approximate 

value of the integral. 

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14 


Divide into equal segments 

as shown in Figure 4. Then 

the width of each segment is: 

The integral I is: 

Figure 4: Multiple (n=4) Segment Trapezoidal Rule 

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15 


The integral I can be broken into h integrals as: 

Applying Trapezoidal rule on each segment gives: 

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16 

Example 2 

The vertical distance covered by a rocket from to seconds is 

given by: 

a) Use two-segment Trapezoidal rule to find the distance covered. 



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17 

Solution 

a) The solution using 2-segment Trapezoidal rule is 

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18 

Then: 


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19 



so the true error is 

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20 


c) The absolute relative true error, , would be 

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Table 1 gives the values 

obtained using multiple 

segment Trapezoidal rule 

for: 

21 


n Value E t 

1 11868 -807 7.296 --- 

2 11266 -205 1.853 5.343 

3 11153 -91.4 0.8265 1.019 

4 11113 -51.5 0.4655 0.3594 

5 11094 -33.0 0.2981 0.1669 

6 11084 -22.9 0.2070 0.09082 

7 11078 -16.8 0.1521 0.05482 

8 11074 -12.9 0.1165 0.03560 

Table 1: Multiple Segment Trapezoidal Rule Values 

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. 

22 

Example 3 

Use Multiple Segment Trapezoidal Rule to find 

the area under the curve 

from 

Using two segments, we get and 

to 

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23 

Then: 

Solution 

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24 


So what is the true value of this integral? 

Making the absolute relative true error: 

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25 


Table 2: Values obtained using Multiple Segment 

Trapezoidal Rule for: 

n Approximate 

Value 

1 0.681 245.91 99.724% 

2 50.535 196.05 79.505% 

4 170.61 75.978 30.812% 

8 227.04 19.546 7.927% 

16 241.70 4.887 1.982% 

32 245.37 1.222 0.495% 

64 246.28 0.305 0.124% 

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26 

Error in Multiple Segment 

Trapezoidal Rule 

The true error for a single segment Trapezoidal rule is given by: 

where is some point in 

What is the error, then in the multiple segment Trapezoidal rule? It will 

be simply the sum of the errors from each segment, where the error in 

each segment is that of the single segment Trapezoidal rule. 

The error in each segment is 

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Similarly: 

It then follows that: 

27 



http:// 


. 

28 



Hence the total error in multiple segment Trapezoidal rule is 

The term 

Hence: 

is an approximate average value of the 

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Below is the table for the integral 

29 



as a function of the number of segments. You can visualize that as the number 

of segments are doubled, the true error gets approximately quartered. 

n Value 

2 11266 -205 1.854 5.343 

4 11113 -51.5 0.4655 0.3594 

8 11074 -12.9 0.1165 0.03560 

16 11065 -3.22 0.02913 0.00401 

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Topic: Simpson’s 1/3 rd Rule 



31 

Basis of Simpson’s 1/3 rd Rule 

Trapezoidal rule was based on approximating the integrand by a first 

order polynomial, and then integrating the polynomial in the interval of 

integration. Simpson’s 1/3rd rule is an extension of Trapezoidal rule 

where the integrand is approximated by a second order polynomial. 

Hence 

Where is a second order polynomial. 

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32 


Choose 

and 

as the three points of the function to evaluate a 0 , a 1 and a 2 . 

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33 


Solving the previous equations for a 0 , a 1 and a 2 give 

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34 

Then 


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35 


Substituting values of a 0 , a 1 , a 2 give 

Since for Simpson’s 1/3rd Rule, the interval [a, b] is broken 

into 2 segments, the segment width 

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Hence 

36 


Because the above form has 1/3 in its formula, it is called Simpson’s 1/3rd Rule. 

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37 

Example 1 

The distance covered by a rocket from t=8 to t=30 is given by 

a) Use Simpson’s 1/3rd Rule to find the approximate value of x 

b) Find the true error, 

c) Find the absolute relative true error, 

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a) 

38 

Solution 

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39 



True Error 

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40 

a)c) Absolute relative true error, 


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41 

Multiple Segment Simpson’s 

1/3rd Rule 

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42 

Multiple Segment Simpson’s 1/3 rd 

Rule 

Just like in multiple segment Trapezoidal Rule, one can subdivide the interval 

[a, b] into n segments and apply Simpson’s 1/3rd Rule repeatedly over 

every two segments. Note that n needs to be even. Divide interval 

[a, b] into equal segments, hence the segment width 

where 

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. 

. 

43 



Apply Simpson’s 1/3rd Rule over each interval, 

f(x) 

. . . 

x 0 x 2 x n-2 x n 

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numericalmethods.eng.usf.edu 

x

Since 

44 



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Then 

45 



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46 



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47 

Example 2 

Use 4-segment Simpson’s 1/3rd Rule to approximate the distance 

covered by a rocket from t= 8 to t=30 as given by 

a) Use four segment Simpson’s 1/3rd Rule to find the approximate 

value of x. 



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a) 

48 

Solution 

Using n segment Simpson’s 1/3rd Rule, 

So 

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49 

Solution (cont.) 

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cont. 

50 


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) 

c) 

51 

In this case, the true error is 

The absolute relative true error 


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52 


Table 1: Values of Simpson’s 1/3rd Rule for Example 2 with multiple segments 

n Approximate Value E t |Є t | 

2 

4 

6 

8 

10 

11065.72 

11061.64 

11061.40 

11061.35 

11061.34 

4.38 

0.30 

0.06 

0.01 

0.00 

0.0396% 

0.0027% 

0.0005% 

0.0001% 

0.0000% 

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53 

Error in the Multiple Segment 

Simpson’s 1/3 rd Rule 

The true error in a single application of Simpson’s 1/3rd Rule is given as 

In Multiple Segment Simpson’s 1/3rd Rule, the error is the sum of the errors 

in each application of Simpson’s 1/3rd Rule. The error in n segment Simpson’s 

1/3rd Rule is given by 

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54 



. 

. 

. 

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55 



Hence, the total error in Multiple Segment Simpson’s 1/3rd Rule is 

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56 



The term is an approximate average value of 

Hence 

where 

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Topic: Romberg Rule 



58 

What is The Romberg Rule? 

Romberg Integration is an extrapolation formula of 

the Trapezoidal Rule for integration. It provides a 

better approximation of the integral by reducing the 

True Error. 

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59 



The true error in a multiple segment Trapezoidal 

Rule with n segments for an integral 

Is given by 

where for each i, is a point somewhere in the 

domain , . 

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60 



The term can be viewed as an 

approximate average value of in . 

This leads us to say that the true error, E t 

previously defined can be approximated as 

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Table 1 shows the results 

obtained for the integral 

using multiple segment 

Trapezoidal rule for 

61 



n Value E t 

1 11868 807 7.296 --- 

2 11266 205 1.854 5.343 

3 11153 91.4 0.8265 1.019 

4 11113 51.5 0.4655 0.3594 

5 11094 33.0 0.2981 0.1669 

6 11084 22.9 0.2070 0.09082 

7 11078 16.8 0.1521 0.05482 

8 11074 12.9 0.1165 0.03560 

Table 1: Multiple Segment Trapezoidal Rule Values 

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62 



The true error gets approximately quartered as 

the number of segments is doubled. This 

information is used to get a better approximation 

of the integral, and is the basis of Richardson’s 

extrapolation. 

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63 

Richardson’s Extrapolation for 


The true error, in the n-segment Trapezoidal rule 

is estimated as 

where C is an approximate constant of 

proportionality. Since 

Where TV = true value and = approx. value 

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64 

Richardson’s Extrapolation for 


From the previous development, it can be shown 

that 

when the segment size is doubled and that 

which is Richardson’s Extrapolation. 

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65 

Example 1 

The vertical distance covered by a rocket from 8 to 30 

seconds is given by 

a) Use Richardson’s rule to find the distance covered. 

Use the 2-segment and 4-segment Trapezoidal 

rule results given in Table 1. 

b) Find the true error, E t for part (a). 


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a) 

66 

Solution 

Using Richardson’s extrapolation formula 

for Trapezoidal rule 

and choosing n=2, 

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Hence 

67 



http:// 


68 


c) The absolute relative true error would then be 

Table 2 shows the Richardson’s extrapolation 

results using 1, 2, 4, 8 segments. Results are 

compared with those of Trapezoidal rule. 

. 

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. 

69 


Table 2: The values obtained using Richardson’s 

extrapolation formula for Trapezoidal rule for 

n Trapezoidal 


1 

2 

4 

8 

11868 

11266 

11113 

11074 

for Trapezoidal 


7.296 

1.854 

0.4655 

0.1165 

Richardson’s 

Extrapolation 

-- 

11065 

11062 

11061 

Table 2: Richardson’s Extrapolation Values 

for Richardson’s 

Extrapolation 

-- 

0.03616 

0.009041 

0.0000 

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70 

Romberg Integration 

Romberg integration is same as Richardson’s 

extrapolation formula as given previously. However, 

Romberg used a recursive algorithm for the 

extrapolation. Recall 

This can alternately be written as 

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Note that the variable TV is replaced by as the 

value obtained using Richardson’s extrapolation formula. 

Note also that the sign is replaced by = sign. 

Hence the estimate of the true value now is 

71 


Where Ch 4 is an approximation of the true error. 

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72 


Determine another integral value with further halving 

the step size (doubling the number of segments), 

It follows from the two previous expressions 

that the true value TV can be written as 

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73 


A general expression for Romberg integration can be 

written as 

The index k represents the order of extrapolation. 

k=1 represents the values obtained from the regular 

Trapezoidal rule, k=2 represents values obtained using the 

true estimate as O(h 2 ). The index j represents the more and 

less accurate estimate of the integral. 

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74 

Example 2 

The vertical distance covered by a rocket from 

to seconds is given by 

Use Romberg’s rule to find the distance covered. Use 

the 1, 2, 4, and 8-segment Trapezoidal rule results as 

given in the Table 1. 

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75 

Solution 

From Table 1, the needed values from original 

Trapezoidal rule are 

where the above four values correspond to using 1, 2, 

4 and 8 segment Trapezoidal rule, respectively. 

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Similarly, 

76 


To get the first order extrapolation values, 

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77 


For the second order extrapolation values, 

Similarly, 

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For the third order extrapolation values, 

Table 3 shows these increased correct values in a tree 

graph. 

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79 


Table 3: Improved estimates of the integral value using Romberg Integration 

1-segment 

2-segment 

4-segment 

8-segment 

11868 

1126 

11113 

11074 

First Order Second Order Third Order 

11065 

11062 

11061 

11062 

11061 

11061 

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Topic: Gauss Quadrature Rule of 




Two-Point Gaussian 

Quadrature Rule 


82 

Basis of the Gaussian 


Previously, the Trapezoidal Rule was developed by the method 

of undetermined coefficients. The result of that development is 

summarized below. 

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83 



The two-point Gauss Quadrature Rule is an extension of the 

Trapezoidal Rule approximation where the arguments of the 

function are not predetermined as a and b but as unknowns 

x 1 and x 2 . In the two-point Gauss Quadrature Rule, the 

integral is approximated as 

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84 



The four unknowns x 1 , x 2 , c 1 and c 2 are found by assuming that 

the formula gives exact results for integrating a general third 

order polynomial, 

Hence 

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It follows that 

85 



Equating Equations the two previous two expressions yield 

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86 



Since the constants a 0 , a 1 , a 2 , a 3 are arbitrary 

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87 

Basis of Gauss Quadrature 

The previous four simultaneous nonlinear Equations have 

only one acceptable solution, 

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88 

Basis of Gauss Quadrature 

Hence Two-Point Gaussian Quadrature Rule 

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Higher Point Gaussian 

Quadrature Formulas 


90 

Higher Point Gaussian Quadrature 

Formulas 

is called the three-point Gauss Quadrature Rule. 

The coefficients c 1 , c 2 , and c 3 , and the functional arguments x 1 , x 2 , and x 3 

are calculated by assuming the formula gives exact expressions for 

integrating a fifth order polynomial 

General n-point rules would approximate the integral 

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91 

Arguments and Weighing Factors 

for n-point Gauss Quadrature 

In handbooks, coefficients and 

arguments given for n-point 

Gauss Quadrature Rule are 

given for integrals 

as shown in Table 1. 

Formulas 

Table 1: Weighting factors c and function 

arguments x used in Gauss Quadrature 

Formulas. 

Points Weighting 

Factors 

2 c 1 = 1.000000000 

c 2 = 1.000000000 

3 c 1 = 0.555555556 

c 2 = 0.888888889 

c 3 = 0.555555556 

4 c 1 = 0.347854845 

c 2 = 0.652145155 

c 3 = 0.652145155 

c 4 = 0.347854845 

Function 

Arguments 

x 1 = -0.577350269 

x 2 = 0.577350269 

x 1 = -0.774596669 

x 2 = 0.000000000 

x 3 = 0.774596669 

x 1 = -0.861136312 

x 2 = -0.339981044 

x 3 = 0.339981044 

x 4 = 0.861136312 

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92 



Points Weighting 

Factors 

Formulas 

Table 1 (cont.) : Weighting factors c and function arguments x used in 

Gauss Quadrature Formulas. 

5 c 1 = 0.236926885 

c 2 = 0.478628670 

c 3 = 0.568888889 

c 4 = 0.478628670 

c 5 = 0.236926885 

6 c 1 = 0.171324492 

c 2 = 0.360761573 

c 3 = 0.467913935 

c 4 = 0.467913935 

c 5 = 0.360761573 

c 6 = 0.171324492 

Function 

Arguments 

x 1 = -0.906179846 

x 2 = -0.538469310 

x 3 = 0.000000000 

x 4 = 0.538469310 

x 5 = 0.906179846 

x 1 = -0.932469514 

x 2 = -0.661209386 

x 3 = -0.2386191860 

x 4 = 0.2386191860 

x 5 = 0.661209386 

x http:// 

6 = 0.932469514 


93 



Formulas 

So if the table is given for integrals, how does one solve 

? The answer lies in that any integral with limits of 

can be converted into an integral with limits Let 

If then 

If then 

Such that: 

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94 



Formulas 

Then Hence 

Substituting our values of x, and dx into the integral gives us 

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95 

Example 1 

For an integral derive the one-point Gaussian Quadrature 

Rule. 

Solution 

The one-point Gaussian Quadrature Rule is 

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96 

Solution 

Assuming the formula gives exact values for integrals 

and 

Since the other equation becomes 

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97 


Therefore, one-point Gauss Quadrature Rule can be expressed as 

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a) 

b) 

c) 

98 

Example 2 

Use two-point Gauss Quadrature Rule to approximate the distance 

covered by a rocket from t=8 to t=30 as given by 

Find the true error, for part (a). 

Also, find the absolute relative true error, for part (a). 

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99 

Solution 

First, change the limits of integration from [8,30] to [-1,1] 

by previous relations as follows 

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. 

. 

100 


Next, get weighting factors and function argument values from Table 1 

for the two point rule, 

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101 


Now we can use the Gauss Quadrature formula 

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since 

102 


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) 

c) 

The true error, , is 

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The absolute relative true error, , is (Exact value = 11061.34m) 

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Trapezoidal Rule Integration

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