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Simpson's Rule

Simpson's rule used for finding out assumed ship's area and volume of hull

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Simpson's Rule

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  1. “Teach A Level Maths”Vol. 1: AS Core Modules 21: Simpson’s Rule © Christine Crisp

  2. Module C3 AQA OCR "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

  3. However, this time, there must be an even number of strips as they are taken in pairs. A very good approximation to a definite integral can be found with Simpson’s rule. As you saw with the Trapezium rule ( and for AQA students with the mid-ordinate rule ), the area under the curve is divided into a number of strips of equal width. I’ll show you briefly how the rule is found but you just need to know the result.

  4. e.g. To estimate we’ll take 4 strips. x x x The rule fits a quadratic curve to the 1st 3 points at the top edge of the strips.

  5. e.g. To estimate we’ll take 4 strips. x x x The rule fits a quadratic curve to the 1st 3 points at the top edge of the strips. Another quadratic curve is fitted to the 3rd, 4th and 5th points.

  6. e.g. To estimate we’ll take 4 strips. x x x The rule fits a quadratic curve to the 1st 3 points at the top edge of the strips. Another quadratic curve is fitted to the 3rd, 4th and 5th points.

  7. e.g. To estimate we’ll take 4 strips. x x x The rule fits a quadratic curve to the 1st 3 points at the top edge of the strips. Another quadratic curve is fitted to the 3rd, 4th and 5th points.

  8. x x x x x The formula for the 1st 2 strips is For the 2nd 2 strips,

  9. In general, We get Notice the symmetry in the formula. The coefficients always end with 4, 1.

  10. SUMMARY where n is the number of strips and must be even. • The width, h, of each strip is given by • Simpson’s rule for estimating an area is ( Notice the symmetry in the formula. ) • The number of ordinates ( y-values ) is odd. • a is the left-hand limit of integration and the 1st value of x. • The accuracy can be improved by increasing n.

  11. e.g. (a) Use Simpson’s rule with 4 strips to estimate giving your answer to 4 d.p. (b) Use your formula book to help you find the exact value of the integral and hence find an approximation for to 3 s.f. Solution: (a) ( It’s a good idea to write down the formula with the correct number of ordinates. Always one more than the number of strips. )

  12. Solution:

  13. Solution: (b) Use your formula book to help you find the exact value of the integral and hence find an approximation for to 3 s.f. (a) The answers to (a) and (b) are approximately equal:

  14. Exercise using Simpson’s rule 1. (a) Estimate with 4 strips, giving your answer to 4 d.p. (b) Find the exact value of the integral and give this correct to 4 d.p. Calculate to 1 s.f. the percentage error in (a).

  15. 1. (a) Estimate using Simpson’s rule with 4 strips, giving your answer to 4 d.p. Solution:

  16. Percentage error (b) Find the exact value of the integral and give this correct to 4 d.p. Calculate to 1 s.f. the percentage error in (a).

  17. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

  18. A very good approximation to a definite integral can be found with Simpson’s rule. As before, the area under the curve is divided into a number of strips of equal width. However, this time, there must be an even number of strips as they are taken in pairs.

  19. SUMMARY • Simpson’s rule for estimating an area is where n is the number of strips and must be even. ( Notice the symmetry in the formula. ) • The number of ordinates ( y-values ) is odd. • The width, h, of each strip is given by • a is the left-hand limit of integration and the 1st value of x. • The accuracy can be improved by increasing n.

  20. e.g. (a) Use Simpson’s rule with 4 strips to estimate giving your answer to 4 d.p. (b) Use your formula book to help you find the exact value of the integral and hence find an approximation for to 3 s.f. Solution: (a) ( It’s a good idea to write down the formula with the correct number of ordinates. Always one more than the number of strips. )

  21. Solution:

  22. Solution: (b) The answers to (a) and (b) are approximately equal:

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