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Numerical Integration

Numerical Integration. Introduction to Numerical Integration. Definitions Upper and Lower Sums Trapezoid Method (Newton-Cotes Methods). Integration. Indefinite Integrals Indefinite Integrals of a function are functions that differ from each other by a constant. Definite Integrals

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Numerical Integration

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  1. Numerical Integration

  2. Introduction to Numerical Integration Definitions Upper and Lower Sums Trapezoid Method (Newton-Cotes Methods)

  3. Integration Indefinite Integrals Indefinite Integrals of a function are functionsthat differ from each other by a constant. Definite Integrals Definite Integrals are numbers.

  4. Fundamental Theorem of Calculus

  5. The Area Under the Curve One interpretation of the definite integral is: Integral = area under the curve f(x) a b

  6. Upper and Lower Sums The interval is divided into subintervals. f(x) a b

  7. Upper and Lower Sums f(x) a b

  8. Example

  9. Example

  10. Upper and Lower Sums • Estimates based on Upper and Lower Sums are easy to obtain for monotonic functions (always increasing or always decreasing). • For non-monotonic functions, finding maximum and minimum of the function can be difficult and other methods can be more attractive.

  11. Newton-Cotes Methods • In Newton-Cote Methods, the function is approximated by a polynomial of order n. • Computing the integral of a polynomial is easy.

  12. Newton-Cotes Methods • Trapezoid Method (First Order Polynomials are used) • Simpson 1/3 Rule (Second Order Polynomials are used)

  13. Trapezoid Method Derivation-One Interval Multiple Application Rule Estimating the Error Recursive Trapezoid Method

  14. Trapezoid Method f(x)

  15. Trapezoid MethodDerivation-One Interval

  16. Trapezoid Method f(x)

  17. Trapezoid MethodMultiple Application Rule f(x) x a b

  18. Trapezoid MethodGeneral Formula and Special Case

  19. Example Given a tabulated values of the velocity of an object. Obtain an estimate of the distance traveled in the interval [0,3]. Distance = integral of the velocity

  20. Example 1

  21. Error in estimating the integralTheorem

  22. Estimating the Error For Trapezoid Method

  23. Example

  24. Example

  25. Example

  26. Recursive Trapezoid Method f(x)

  27. Recursive Trapezoid Method f(x) Based on previous estimate Based on new point

  28. Recursive Trapezoid Method f(x) Based on previous estimate Based on new points

  29. Recursive Trapezoid MethodFormulas

  30. Recursive Trapezoid Method

  31. Example on Recursive Trapezoid Estimated Error = |R(3,0) – R(2,0)| = 0.009669

  32. Advantages of Recursive Trapezoid Recursive Trapezoid: • Gives the same answer as the standard Trapezoid method. • Makes use of the available information to reduce the computation time. • Useful if the number of iterations is not known in advance.

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