# 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:
• 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

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