21: simpson’s rule © christine crisp “teach a level maths” vol. 1: as core modules
TRANSCRIPT
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
As you saw with the Trapezium 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.
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 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
The formula for the 1st 2 strips is
)4(3 210 yyyh
x 0y
x 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...24243
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.
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. )
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 424
3yyyyy
h
50640809411801 y
) d.p. ( 478540
1
021
1dx
x 43210 424
3
250yyyyy
Simpson’s Rule
Solution:
1
021
1dx
x 1
01tan x
0tan1tan 11 4
) d.p. ( 478540
1
021
1dx
x(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 Rule
Exercise
3
1
1dx
xusing 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
yyyyyh
A Solutio
n:
using Simpson’s rule
with 4 strips, giving your answer to 4 d.p.
1. (a) Estimate3
1
1dx
x
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
1ln
1xdx
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.
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 424
3yyyyy
h
50640809411801 y
) d.p. ( 478540
1
021
1dx
x 43210 424
3
250yyyyy
Simpson’s RuleSolutio
n:(b)
1
021
1dx
x 1
01tan x
0tan1tan 11
4
The answers to (a) and (b) are approximately equal:
785404
So,
) s.f. 3( 143