Upload
arvin-loui-bascon
View
219
Download
0
Embed Size (px)
Citation preview
7/23/2019 Approximation Rule
1/4
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.php7/23/2019 Approximation Rule
2/4
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/395207/23/2019 Approximation Rule
3/4
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/7/23/2019 Approximation Rule
4/4
http://www.slideshare.net/
http://www.slideshare.net/http://www.slideshare.net/