08 numerical integration

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Text of 08 numerical integration

  • 1. Numerical Integration
  • 2. Objectives
    • The student should be able to
      • Understand the need for numerical integration
      • Derive the trapezoidal rule using linear interpolation
      • Apply the trapezoidal rule
      • Derive Simpsons rule using parabolic interpolation
      • Apply Simpsons rule
  • 3. Need for Numerical Integration!
  • 4. Interpolation!
    • If we have a function that needs to be integrated between two points
    • We may use an approximate form of the function to integrate!
    • Polynomials are always integrable
    • Why dont we use a polynomial to approximate the function, then evaluate the integral
  • 5. Example
    • To perform the definite integration of the function between (x 0 & x 1 ), we may interpolate the function between the two points as a line.
  • 6. Example
    • Performing the integration on the approximate function:
  • 7. Example
    • Performing the integration on the approximate function:
    • Which is equivalent to the area of the trapezium!
  • 8. The Trapezoidal Rule Integrating from x 0 to x 2 :
  • 9. General Trapezoidal Rule
    • For all the points equally separated (x i+1 -x i =h)
    • We may write the equation of the previous slide:
  • 10. In general Where n is the number if intervals and h=total interval/n
  • 11. Example
    • Integrate
    • Using the trapezoidal rule
    • Use 2 points and compare with the result using 3 points
  • 12. Solution
    • Using 2 points (n=1), h=(1-0)/(1)=1
    • Substituting:
  • 13. Solution
    • Using 3 points (n=2), h=(1-0)/(2)=0.5
    • Substituting:
  • 14. Quadratic Interpolation
    • If we get to interpolate a quadratic equation between every neighboring 3 points, we may use Newtons interpolation formula:
  • 15. Integrating
  • 16. After substitutions and manipulation!
  • 17. For 4-Intervals
  • 18. In General: Simpsons Rule NOTE: the number of intervals HAS TO BE even
  • 19. Example
    • Integrate
    • Using the Simpson rule
    • Use 3 points
  • 20. Solution
    • Using 3 points (n=2), h=(1-0)/(2)=0.5
    • Substituting:
    • Which is the exact solution!
  • 21. Homework #7
    • Chapter 21, pp. 610-612, numbers: 21.1, 21.3, 21.5, 21.25, 21.28.
    • Due date: Week 8-12 May 2005