Simpson

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Simpsons 1/3rd Rule of Integration

• Mechanical Engineering Majors
• Authors Autar Kaw, Charlie Barker
• http//numericalmethods.eng.usf.edu
• Transforming Numerical Methods Education for STEM
  Undergraduates

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Simpsons 1/3rd Rule of Integrationhttp//nume
ricalmethods.eng.usf.edu
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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
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• Simpsons 1/3rd Rule

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Basis of Simpsons 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. Simpsons 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 Simpsons 1/3rd Rule
Choose
and
as the three points of the function to evaluate
a0, a1 and a2.
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Basis of Simpsons 1/3rd Rule
Solving the previous equations for a0, a1 and a2
give
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Basis of Simpsons 1/3rd Rule
Then
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Basis of Simpsons 1/3rd Rule
Substituting values of a0, a1, a 2 give
Since for Simpsons 1/3rd Rule, the interval a,
b is broken
into 2 segments, the segment width
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Basis of Simpsons 1/3rd Rule
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Example 1

• A trunnion of diameter 12.363 has to be cooled
  from a room temperature of 80oF before it is
  shrink fit into a steel hub (Figure 2).
• The equation that gives the diametric
  contraction of the trunnion in dry-ice/alcohol
  (boiling temperature is -108oF) is given by

Figure 2. Trunnion to be slided through the hub
after contracting.

1. Use Simpsons 1/3rd Rule to find the contraction.
2. Find the true error, for part (a).
3. Find the absolute relative true error, for
   part (a).

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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)
c) Absolute relative true error,
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• Multiple Segment Simpsons 1/3rd Rule

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Multiple Segment Simpsons 1/3rd Rule
Just like in multiple segment Trapezoidal Rule,
one can subdivide the interval
a, b into n segments and apply Simpsons 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 Simpsons 1/3rd Rule
Apply Simpsons 1/3rd Rule over each interval,
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Multiple Segment Simpsons 1/3rd Rule
Since
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Multiple Segment Simpsons 1/3rd Rule
Then
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Multiple Segment Simpsons 1/3rd Rule
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Example 2
A trunnion of diameter 12.363 has to be cooled
from a room temperature of 80oF before it is
shrink fit into a steel hub (Figure 2). The
equation that gives the diametric contraction of
the trunnion in dry-ice/alcohol (boiling
temperature is -108oF) is given by
a) Use 4-segment Simpsons 1/3rd Rule to find the
contraction. 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 Simpsons 1/3rd Rule,
a)
So
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Solution (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 Simpsons 1/3rd Rule for
Example 2 with multiple segments
Approximate Value
2 4 6 8 10 -0.013689 -0.013689 -0.013689 -0.013689 -0.013689 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
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Error in the Multiple Segment Simpsons 1/3rd Rule
The true error in a single application of
Simpsons 1/3rd Rule is given as
In Multiple Segment Simpsons 1/3rd Rule, the
error is the sum of the errors
in each application of Simpsons 1/3rd Rule. The
error in n segment Simpsons
1/3rd Rule is given by
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Error in the Multiple Segment Simpsons 1/3rd Rule
. . .
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Error in the Multiple Segment Simpsons 1/3rd Rule
Hence, the total error in Multiple Segment
Simpsons 1/3rd Rule is
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Error in the Multiple Segment Simpsons 1/3rd Rule
The term
is an approximate average value of
Hence
where
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Additional Resources

• For all resources on this topic such as digital
  audiovisual lectures, primers, textbook chapters,
  multiple-choice tests, worksheets in MATLAB,
  MATHEMATICA, MathCad and MAPLE, blogs, related
  physical problems, please visit
• http//numericalmethods.eng.usf.edu/topics/simpso
  ns_13rd_rule.html

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• THE END
• http//numericalmethods.eng.usf.edu