Integration. Topic: Trapezoidal Rule. Major: General Engineering. Author: Autar Kaw, Charlie Barker.

Integration

Topic: Trapezoidal Rule

Major: General Engineering

Author: Autar Kaw, Charlie Barker

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What is Integration

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

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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…

where

and

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Basis of Trapezoidal Rule

Then the integral of that function is approximated by the integral of that n

order polynomial.

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

under the linear polynomial,

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Derivation of the Trapezoidal Rule

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Method Derived From Geometry

The area under the curve is a trapezoid.

The integral

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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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Solution

a)

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Solution (cont)

a)

b) The exact value of the above integral is

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Solution (cont)

b)

c) The absolute relative true error , , would be

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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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Multiple Segment Trapezoidal Rule

With

Hence:

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Multiple Segment Trapezoidal Rule

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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Multiple Segment Trapezoidal Rule

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

Divide into equal segments as shown in Figure 4. Then the width of each segment is:

The integral I is:

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Multiple Segment Trapezoidal Rule

The integral I can be broken into h integrals as:

Applying Trapezoidal rule on each segment gives:

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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.

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

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

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Solution

a) The solution using 2-segment Trapezoidal rule is

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Solution (cont)

Then:

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Solution (cont)

b) The exact value of the above integral is

so the true error is

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Solution (cont)

c) The absolute relative true error , , would be

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Solution (cont)

Table 1 gives the values obtained using multiple segment Trapezoidal rule for:

n Value E

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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Example 3

Use Multiple Segment Trapezoidal Rule to find the area under the curve

from to

.

Using two segments, we get _and

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Solution

Then:

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Solution (cont)

So what is the true value of this integral?

Making the absolute relative true error:

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Solution (cont)

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%

Table 2: Values obtained using Multiple Segment

Trapezoidal Rule for:

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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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Error in Multiple Segment Trapezoidal Rule

Similarly:

It then follows that:

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Error in Multiple Segment Trapezoidal Rule

Hence the total error in multiple segment Trapezoidal rule is

The term is an approximate average value of the

.

Hence:

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Error in Multiple Segment Trapezoidal Rule

Below is the table for the integral

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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Integration

Topic: Simpson’s 1/3 ^rd Rule

Major: General Engineering

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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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Basis of Simpson’s 1/3 ^rd Rule

Choose

and

as the three points of the function to evaluate a

, a

and a

.

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Basis of Simpson’s 1/3 ^rd Rule

Solving the previous equations for a

, a

and a

give

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Basis of Simpson’s 1/3 ^rd Rule

Then

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Basis of Simpson’s 1/3 ^rd Rule

Substituting values of a

, a

, a

give

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

into 2 segments, the segment width

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Basis of Simpson’s 1/3 ^rd Rule

Hence

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

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Example 1

a) Use Simpson’s 1/3rd Rule to find the approximate value of x The distance covered by a rocket from t=8 to t=30 is given by

b) Find the true error,

c) Find the absolute relative true error,

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Solution

a)

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Solution (cont)

b) The exact value of the above integral is

True Error

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Solution (cont)

a)

c) Absolute relative true error,

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Multiple Segment Simpson’s

1/3rd Rule

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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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Multiple Segment Simpson’s 1/3 ^rd Rule

.

.

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

f(x)

. . .

x₀ x₂ x_n-2 x_n

x

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Multiple Segment Simpson’s 1/3 ^rd Rule

Since

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Multiple Segment Simpson’s 1/3 ^rd Rule

Then

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Multiple Segment Simpson’s 1/3 ^rd Rule

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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.

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

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

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Solution

Using n segment Simpson’s 1/3rd Rule,

So

a)

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Solution (cont.)

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Solution (cont.)

cont.

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Solution (cont.)

In this case, the true error is

b)

The absolute relative true error

c)

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Solution (cont.)

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

n Approximate Value E

|Є

| 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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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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Error in the Multiple Segment Simpson’s 1/3 ^rd Rule

.

.

.

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Error in the Multiple Segment Simpson’s 1/3 ^rd Rule

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

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Error in the Multiple Segment Simpson’s 1/3 ^rd Rule

The term is an approximate average value of

Hence

where

Integration

Topic: Romberg Rule

Major: General Engineering

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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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Error in Multiple Segment Trapezoidal Rule

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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Error in Multiple Segment Trapezoidal Rule

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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Error in Multiple Segment Trapezoidal Rule

Table 1 shows the results obtained for the integral using multiple segment Trapezoidal rule for

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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Error in Multiple Segment Trapezoidal Rule

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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Richardson’s Extrapolation for Trapezoidal Rule

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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Richardson’s Extrapolation for Trapezoidal Rule

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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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).

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

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Solution

a)

Using Richardson’s extrapolation formula for Trapezoidal rule

and choosing n=2,

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Solution (cont.)

b) The exact value of the above integral is

Hence

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Solution (cont.)

.

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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.

Solution (cont.)

Table 2: The values obtained using Richardson’s extrapolation formula for Trapezoidal rule for

n Trapezoidal

Rule for Trapezoidal

Rule Richardson’s

Extrapolation for Richardson’s Extrapolation 1

2 4 8

11868 11266 11113 11074

7.296 1.854 0.4655 0.1165

-- 11065 11062 11061

-- 0.03616 0.009041

0.0000 Table 2: Richardson’s Extrapolation Values

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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.

Romberg Integration

Hence the estimate of the true value now is

Where Ch ⁴ is an approximation of the true error.

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Romberg Integration

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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Romberg Integration

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.

A general expression for Romberg integration can be

written as

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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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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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Solution (cont.)

To get the first order extrapolation values,

Similarly,

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Solution (cont.)

For the second order extrapolation values,

Similarly,

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Solution (cont.)

For the third order extrapolation values,

Table 3 shows these increased correct values in a tree

graph.

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Solution (cont.)

11868 1126 11113 11074

11065 11062 11061

11062 11061

11061 1-segment

2-segment 4-segment 8-segment

First Order Second Order Third Order

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

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Integration

Topic: Gauss Quadrature Rule of Integration

Major: General Engineering

Two-Point Gaussian

Quadrature Rule

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Basis of the Gaussian Quadrature Rule

Previously, the Trapezoidal Rule was developed by the method

of undetermined coefficients. The result of that development is

summarized below.

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Basis of the Gaussian Quadrature Rule

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

and x

. In the two-point Gauss Quadrature Rule, the

integral is approximated as

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Basis of the Gaussian Quadrature Rule

The four unknowns x

, x

, c

and c

are found by assuming that the formula gives exact results for integrating a general third order polynomial,

Hence

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Basis of the Gaussian Quadrature Rule

It follows that

Equating Equations the two previous two expressions yield

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Basis of the Gaussian Quadrature Rule

Since the constants a

, a

, a

, a

are arbitrary

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Basis of Gauss Quadrature

The previous four simultaneous nonlinear Equations have

only one acceptable solution,

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Basis of Gauss Quadrature

Hence Two-Point Gaussian Quadrature Rule

Higher Point Gaussian

Quadrature Formulas

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Higher Point Gaussian Quadrature Formulas

is called the three-point Gauss Quadrature Rule.

The coefficients c

, c

, and c

, and the functional arguments x

, x

, and x

are calculated by assuming the formula gives exact expressions for

General n-point rules would approximate the integral

integrating a fifth order polynomial

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Arguments and Weighing Factors for n-point Gauss Quadrature

Formulas

In handbooks, coefficients and Gauss Quadrature Rule are

as shown in Table 1.

Points Weighting

Factors Function

Arguments 2 c₁ = 1.000000000

c₂ = 1.000000000

x₁ = -0.577350269 x₂ = 0.577350269 3 c₁ = 0.555555556

c₂ = 0.888888889 c₃ = 0.555555556

x₁ = -0.774596669 x₂ = 0.000000000 x₃ = 0.774596669 4 c₁ = 0.347854845

c₂ = 0.652145155 c₃ = 0.652145155 c₄ = 0.347854845

x₁ = -0.861136312 x₂ = -0.339981044 x₃ = 0.339981044 x₄ = 0.861136312

arguments given for n-point given for integrals

Table 1: Weighting factors c and function

arguments x used in Gauss Quadrature Formulas.

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Arguments and Weighing Factors for n-point Gauss Quadrature

Formulas

Points Weighting

Factors Function

Arguments 5 c₁ = 0.236926885

c₂ = 0.478628670 c₃ = 0.568888889 c₄ = 0.478628670 c₅ = 0.236926885

x₁ = -0.906179846 x₂ = -0.538469310 x₃ = 0.000000000 x₄ = 0.538469310 x₅ = 0.906179846 6 c₁ = 0.171324492

c₂ = 0.360761573 c₃ = 0.467913935 c₄ = 0.467913935 c₅ = 0.360761573 c₆ = 0.171324492

x₁ = -0.932469514 x₂ = -0.661209386 x₃ = -0.2386191860 x₄ = 0.2386191860 x₅ = 0.661209386 x₆ = 0.932469514

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

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Arguments and Weighing Factors for n-point Gauss Quadrature

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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Arguments and Weighing Factors for n-point Gauss Quadrature

Formulas

Then Hence

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

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Example 1

For an integral derive the one-point Gaussian Quadrature Rule.

Solution

The one-point Gaussian Quadrature Rule is

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Solution

Assuming the formula gives exact values for integrals and

Since the other equation becomes

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Solution (cont.)

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

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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).

a)

b)

c)

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Solution

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

by previous relations as follows

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Solution (cont)

.

Next, get weighting factors and function argument values from Table 1 for the two point rule,

.

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Solution (cont.)

Now we can use the Gauss Quadrature formula

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Solution (cont)

since

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Solution (cont)

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

The true error, , is b)