Numerical Integration

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Numerical Integration

• Lesson 6.5

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News from Space

• A new species has been trapped the rare zoid
• Math studentshave long known ofefforts of
  "trapezoid" expeditions
• Trapezoids were so effective, zoids were thought
  to be extinct!

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Trapezoidal Rule

• Instead of calculatingapproximation
  rectangleswe will use trapezoids
• More accuracy
• Area of a trapezoid

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Trapezoidal Rule

• Trapezoidal rule approximates the integral
• Calculator function for f(x)(?(2f(ak(b-a)/n),
  k,1,n-1)f(a)f(b))(b-a)/(n2)?trap(a,b,n)

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Trapezoidal Rule

• Entering the trapezoidal rule into the calculator
• f(x) must be defined for this to work

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Trapezoidal Rule

• Try using the trapezoidal
  rule
• Check with integration

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Simpson's Rule

• As before, we dividethe interval into n parts
• n must be even
• Instead of straight lines wedraw parabolas
  through each group of three consecutive points
• This approximates the original curve for finding
  definite integral formula shown below

Snidly Fizbane Simpson

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Simpson's Rule

• Our calculator can do this for us also
• The function is more than a one liner
• We will use the program editor
• Choose APPS,7Program Editor3New
• Specify Function,name it simp

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Simpson's Rule

• Enter the parameters a, b, and n between the
  parentheses

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Simpson's Rule

• Specify a function for f(x)
• When you call simp(a,b,n),
• Make sure n is an even number
• Note the accuracy of the approximation

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Using Data

• Given table of data, use trapezoidal rule to
  determine area under the curve
• dx ?

x 2.00 2.10 2.20 2.30 2.40 2.50 2.60
y 4.32 4.57 5.14 5.78 6.84 6.62 6.51
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Using Data

• Given table of data, use Simpson's rule to
  determine area under the curve

x 2.00 2.10 2.20 2.30 2.40 2.50 2.60
y 4.32 4.57 5.14 5.78 6.84 6.62 6.51
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Assignment

• Lesson 6.5
• Page 250
• Exercises 1 21 odd