21: Simpson’s Rule21: Simpson’s Rule

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”Vol. 1: AS Core Vol. 1: AS Core

ModulesModules

Simpson’s Rule

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Module C3AQA

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

2xey

e.g. To estimate we’ll take 4 strips. 2

0

2dxe x

The rule fits a quadratic curve to the 1st 3 points at the top edge of the strips.

x

x

x

Simpson’s Rule

2xey

x

x

x

e.g. To estimate we’ll take 4 strips. 2

0

2dxe 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.

Simpson’s Rule

2xey x

e.g. To estimate we’ll take 4 strips. 2

0

2dxe 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.

xx

Simpson’s Rule

2xey xx

x

e.g. To estimate we’ll take 4 strips. 2

0

2dxe 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.

Simpson’s Rule

2xey

The formula for the 1st 2 strips is)4(

3 210 yyyh

x 0yx 1y

h

For the 2nd 2 strips, )4(

3 432 yyyh

x 3y4y

x

2yx

Simpson’s Rule

Notice the symmetry in the formula.The coefficients always end with 4, 1.

)4(3 210 yyyh

We get

)4(3 432 yyyh

)424(3 43210 yyyyyh

In general,

b

a

dxy

nnn yyyyyyyyh 1243210 42...2424

3

Simpson’s RuleSUMMARY

where n is the number of strips and must be even.

nab

h

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 1st value of x.

nnn

b

a

yyyyyyyyhydx 1243210 42...24243

The number of ordinates ( y-values ) is odd.

( Notice the symmetry in the formula. )

Simpson’s Rule

1

021

1 dxx

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) 43210 424

3yyyyyhA

( 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

1750502500 x

Solution:

2504

01,4

hn

1

021

1 dxx

43210 4243

yyyyyh

50640809411801 y

) d.p. ( 478540

1

021

1 dxx

43210 4243250 yyyyy

Simpson’s Rule

Solution:

1

021

1 dxx

10

1tan x

0tan1tan 11 4

) d.p. ( 478540

1

021

1 dxx

(a)

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

785404

So,

785404 ) s.f. 3( 143

(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 RuleExercise

3

1

1 dxx

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

43210 4243

yyyyyhA Solutio

n:

using Simpson’s rule

with 4 strips, giving your answer to 4 d.p.

1. (a) Estimate3

1

1 dxx

504

13,4

hn

3522511 x33333040506666701 y

) d.p. 4(1000113

1 dx

x

Simpson’s Rule

) d.p. 4((a) 1000113

1 dx

x

31

3

1ln1 xdx

x

(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).

1ln3ln 3ln

) d.p. 4(09861Percentage

error 10009861

0986110001

) s.f. 1( %10

Simpson’s Rule

Simpson’s Rule

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.

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 RuleSUMMARY

where n is the number of strips and must be even.

nab

h

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.

nnn

b

a

yyyyyyyyhydx 1243210 42...24243

The number of ordinates ( y-values ) is odd.

( Notice the symmetry in the formula. )

a is the left-hand limit of integration and the 1st value of x.

Simpson’s Rule

1

021

1 dxx

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) 43210 424

3yyyyyhA

( 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

1750502500 x

Solution:

2504

01,4

hn

1

021

1 dxx

43210 4243

yyyyyh

50640809411801 y

) d.p. ( 478540

1

021

1 dxx

43210 4243250 yyyyy

Simpson’s RuleSolutio

n:(b)

1

021

1 dxx

10

1tan x

0tan1tan 11

4

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

785404

So,

) s.f. 3( 143