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Integration Topic: Simpson’s 1/3 rd Rule Major: Computer Engineering f(x) y a b x What is Integration? Integration The process of measuring the area under a curve. Where: f(x) is the integrand a= lower limit of integration b= upper limit of integration Simpson’s 1/3 rd Rule

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Integration l.jpg

Integration

Topic: Simpson’s 1/3rd Rule

Major: Computer Engineering

http://numericalmethods.eng.usf.edu


What is integration l.jpg

f(x)

y

a

b

x

What is Integration?

Integration

The process of measuring the area under a curve.

Where:

f(x) is the integrand

a= lower limit of integration

b= upper limit of integration

http://numericalmethods.eng.usf.edu


Slide3 l.jpg

Simpson’s 1/3rd Rule

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Basis of simpson s 1 3 rd rule l.jpg

Where is a second order polynomial.

Basis of Simpson’s 1/3rd 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

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Basis of simpson s 1 3 rd rule5 l.jpg

Basis of Simpson’s 1/3rd Rule

Choose

and

as the three points of the function to evaluate a0, a1 and a2.

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Basis of simpson s 1 3 rd rule6 l.jpg

Basis of Simpson’s 1/3rd Rule

Solving the previous equations for a0, a1 and a2 give

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Basis of simpson s 1 3 rd rule7 l.jpg

Basis of Simpson’s 1/3rd Rule

Then

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Basis of simpson s 1 3 rd rule8 l.jpg

Basis of Simpson’s 1/3rd Rule

Substituting values of a0, a1, 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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Basis of simpson s 1 3 rd rule9 l.jpg

Hence

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

Basis of Simpson’s 1/3rd Rule

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Example 1 l.jpg

Example 1

Human vision has the remarkable ability to infer 3D shapes from 2D

images. The intriguing question is: can we replicate some of these

abilities on a computer? Yes, it can be done and to do this, integration

of vector fields is required. The following integral needs to integrated.

where

  • Use single segment Trapezoidal rule to find the distance covered.

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

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

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Solution l.jpg

Solution

a)

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Solution cont l.jpg

Solution (cont)

b) The exact value of the above integral is found using Maple

for calculating the true error and relative true error.

True Error

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Solution cont13 l.jpg

Solution (cont)

  • c) Absolute relative true error,

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Slide14 l.jpg

Multiple Segment Simpson’s 1/3rd Rule

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Multiple segment simpson s 1 3 rd rule l.jpg

Multiple Segment Simpson’s 1/3rd 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 rule16 l.jpg

f(x)

. . .

x

x0

x2

xn-2

xn

Multiple Segment Simpson’s 1/3rd Rule

.

.

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

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Multiple segment simpson s 1 3 rd rule17 l.jpg

Multiple Segment Simpson’s 1/3rd Rule

Since

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Multiple segment simpson s 1 3 rd rule18 l.jpg

Multiple Segment Simpson’s 1/3rd Rule

Then

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Multiple segment simpson s 1 3 rd rule19 l.jpg

Multiple Segment Simpson’s 1/3rd Rule

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Example 2 l.jpg

Example 2

Human vision has the remarkable ability to infer 3D shapes from 2D

images. The intriguing question is: can we replicate some of these

abilities on a computer? Yes, it can be done and to do this, integration

of vector fields is required. The following integral needs to integrated.

  • 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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Solution21 l.jpg

Solution

a)

Using n segment Simpson’s 1/3rd Rule,

So

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Solution cont22 l.jpg

Solution (cont.)

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Solution cont23 l.jpg

Solution (cont.)

cont.

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Solution cont24 l.jpg

Solution (cont.)

b)

In this case, the true error is

c)

The absolute relative true error

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Solution cont25 l.jpg

Solution (cont.)

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

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Error in the multiple segment simpson s 1 3 rd rule l.jpg

Error in the Multiple Segment Simpson’s 1/3rd 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 rule27 l.jpg

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

.

.

.

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Error in the multiple segment simpson s 1 3 rd rule28 l.jpg

Error in the Multiple Segment Simpson’s 1/3rd 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 rule29 l.jpg

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

The term

is an approximate average value of

Hence

where

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