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

Simpson’s Rule "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" Module C3 AQA OCR

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. A very good approximation to a definite integral can be found with Simpson’s rule. However, this time, there must be an even number of strips as they are taken in pairs. I’ll show you briefly how the rule is found but you just need to know the result.

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

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

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

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

Simpson’s Rule The formula for the 1 st 2 strips is x x For the 2 nd 2 strips, x x x

Simpson’s Rule Notice the symmetry in the formula. The coefficients always end with 4, 1. We get In general,

Simpson’s Rule 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  The accuracy can be improved by increasing n.  a is the left-hand limit of integration and the 1 st value of x.  The number of ordinates ( y -values ) is odd. ( Notice the symmetry in the formula. )

Simpson’s Rule 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. )

Simpson’s Rule Solution:

Simpson’s Rule Solution: (a) The answers to (a) and (b) are approximately equal: (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.

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

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

Simpson’s Rule (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). Percentage error

Simpson’s Rule

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

Simpson’s Rule 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  The accuracy can be improved by increasing n.  The number of ordinates ( y -values ) is odd. ( Notice the symmetry in the formula. )  a is the left-hand limit of integration and the 1 st value of x.

Simpson’s Rule 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. )

Simpson’s Rule Solution:

Simpson’s Rule Solution: (b) The answers to (a) and (b) are approximately equal: