Numerical Integration

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Numerical Integration

• Dr. Asaf Varol
• avarol_at_mix.wvu.edu

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Numerical Integration

• Numerical integration is a primary tool used for
  definite integrals that cannot be solved
  analytically. A numerical integration rule has
  the form
• we investigate several basic quadrature formulas
  that use function values at equally spaced
  points these methods are known as Newton-Cotes
  formulas. There are two types of Newton Cotes
  formulas, depending on whether or not the
  function values at the ends of the interval of
  integration are used. The trapezoid and Simpson
  rules are examples of closed formulas, in which
  the endpoint values are used. The midpoint rule
  is the simplest example of an open formula, in
  which the endpoints are not used 2.

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Newton-Cotes Closed FormulasTrapezoid Rule

• One of the simplest ways to approximate the area
  under a curve is to approximate the curve by a
  straight line. The trapezoid rule approximates
  the curve by the straight line that passes
  through the points a, f(a) and b, f(b), the two
  ends of the interval of interest. We have x0a,
  x1b, and hb-a, and then

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Example
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Matlab Program
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Diagram
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Newton-Cotes Closed FormulasSimpsons Rule
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Newton-Cotes Closed FormulasSimpsons Rule
(Contd)
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Matlab Program
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Diagram
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Newton-Cotes Closed FormulasMidpoint Rule
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Figure

• Figure given on the right side compares the
  actual value of the area with that found by using
  the midpoint rule. The area given by the integral
  S (hatched) and the approximation using the
  midpoint rule (shaded) 2.

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Matlab Program
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Diagram
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Gaussian quadrature
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Gaussian quadrature
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Example
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End of Chapter 5
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References

• Celik, Ismail, B., Introductory Numerical
  Methods for Engineering Applications, Ararat
  Books Publishing, LCC., Morgantown, 2001
• Fausett, Laurene, V. Numerical Methods,
  Algorithms and Applications, Prentice Hall, 2003
  by Pearson Education, Inc., Upper Saddle River,
  NJ 07458
• Rao, Singiresu, S., Applied Numerical Methods
  for Engineers and Scientists, 2002 Prentice Hall,
  Upper Saddle River, NJ 07458
• Mathews, John, H. Fink, Kurtis, D., Numerical
  Methods Using MATLAB Fourth Edition, 2004
  Prentice Hall, Upper Saddle River, NJ 07458
• Varol, A., Sayisal Analiz (Numerical Analysis),
  in Turkish, Course notes, Firat University, 2001