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TRANSCRIPT
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. )
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
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