Simpson’s Rule Mini Whiteboards To check your understanding of the video.

Simpson’s Rule

Mini Whiteboards

To check your understanding of

the video

Now work out the values of x0, x1, x2, x3, x4, x5, x6, x7, x8.

Simpson’s Rule

Objectives:To recognise and apply Simpson’s

rule to approximate areas bounded by

curves.

SUMMARY

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

n

abh

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

yyyyyyyyh

ydx 1243210 42...24243

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

( Notice the symmetry in the formula. )

Exam Questions

Questions about the integral ∫02 √(1+x3)dx. The value of this

integral, correct to four decimal places, is 3.2413.

Simpson’s rule gives a value of 3.2396 therefore the percentage error is: (a) -0.0525% (b) -0.0524% (c) 0.0524% (d) 0.0525%

Integration on GDC

Multi guess worksheet

You should be ok with questions 1 to 7

Exercise DPage 142

Extension:Mixed questions

Page 143

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.

n

abh

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

yyyyyyyyh

ydx 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

1dx

x

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

3yyyyy

hA

( 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

1dx

x 43210 4243

yyyyyh

50640809411801 y

) d.p. ( 478540

1

021

1dx

x 43210 424

3

250yyyyy

Simpson’s RuleSolutio

n:(b)

1

021

1dx

x 101tan x

0tan1tan 11

4

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

785404

So,

) s.f. 3( 143

Extension1

2

3

Extension1

2

3