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Approximation Rules

Simpsons Rule - Simpson's rule is used to find the approximate area under agraph. Since integration between two limits also gives the area under a graph then

Simpson's rule can often be used as a way of finding an approximate value of a

definite integral.It is an improvement on the trapezium ruleas it uses a parabola

rather than a straight line between intervals as an approximation to the curve xo

a

b

f(x )dx x3 [ f(x0 )+4 f(x1)+2 f(x2 )++2 f(xn2)+4 f(xn1 )+ f(xn)]

xample!

Approximating the ff. Integral using n " # and the Simpsons Rule

0

2

ex

2

dx

Solution!

$idth of subintervals!

x=2

0

4=1

2

0

2

ex

2

dx=1/23

[e (0 )2

+4 (e )(.5 )2

+2 (e )(1 )2

+4 (e )(1.5 )2

+e (2 )2

]
http://www.examsolutions.net/maths-revision/core-maths/integration/definite/tutorial-1.phphttp://www.examsolutions.net/maths-revision/core-maths/numerical-methods/integration/trapezium-rule/tutorial-1.phphttp://www.examsolutions.net/maths-revision/core-maths/numerical-methods/integration/trapezium-rule/tutorial-1.phphttp://www.examsolutions.net/maths-revision/core-maths/integration/definite/tutorial-1.php


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Is %ual to!

" &.()(#

*rape+oidal Rule - *he trapezoidal ruleis a method for approximating a definiteintegralby evaluating the integrandat twopoints. *he formal rule is given by

a

b

f(x ) dx h2

[ f( a )+ f(b)]

where h"b-a.

*his rule comes from determining the areaof a right

trape+oidwithbasesof lengthsf,a and f,b respectively and a heightof length h.

$hen using a graphto illustrate the trape+oidal rule the heightof the right

trape+oid is actually hori+ontal and thebasesare vertical.

xample!

Approximating the Integral using *rape+oidal Rule

0

2

ex2

dx

Solution

x=204

=1

2

0

2

ex

2

dx=1/22

[e (0 )2

+2 (e )(.5 )2

+2 (e )(1 )2

+2 (e )(1.5 )2

+e (2 )2

]
http://planetmath.org/node/34318http://planetmath.org/node/31637http://planetmath.org/node/31637http://planetmath.org/node/40599http://planetmath.org/node/38173http://planetmath.org/node/36778http://planetmath.org/node/39518http://planetmath.org/node/39518http://planetmath.org/Base9http://planetmath.org/node/39617http://planetmath.org/Height6http://planetmath.org/node/33701http://planetmath.org/node/36739http://planetmath.org/node/39520http://planetmath.org/node/34318http://planetmath.org/node/31637http://planetmath.org/node/31637http://planetmath.org/node/40599http://planetmath.org/node/38173http://planetmath.org/node/36778http://planetmath.org/node/39518http://planetmath.org/node/39518http://planetmath.org/Base9http://planetmath.org/node/39617http://planetmath.org/Height6http://planetmath.org/node/33701http://planetmath.org/node/36739http://planetmath.org/node/39520


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Is %ual to !

" /0.#11)

2urands Rule , *he 3ewton 4otes 5ormula

x

1

x n

f(x ) dx h[ 25f1+ 1110 f2

+ f3++ fn2+11

10fn1+

2

5fn]

$eddles Rule , *he 3ewton 4otes 5ormula 6 n"#

Is a method of Integration . *he 3ewton 4otes 5ormula with 3"#

a

b

f(x )dx= 310

h [ f0+5 f1+ f2+6 f3+ f4+5 f5+ f6 ]

$here! n " #

h "ba6

References:

http://www.examsolution

http://planetmath.org/s.net/

http://tutorial.math.lamar.edu/

http://archive.lib.msu.edu/
http://www.examsolution/http://planetmath.org/s.net/http://tutorial.math.lamar.edu/http://archive.lib.msu.edu/http://www.examsolution/http://planetmath.org/s.net/http://tutorial.math.lamar.edu/http://archive.lib.msu.edu/


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