Derivation of Simpson's 1/3 rule

Derivation of Simpson's 1/3 rule

1. Hello!
1

2. Mehedi Hasan
2013-2-50-003
2

3. Simpson’s 1/3
rd
Rule of
Integration

4. What is Integration?
ò
=
b
a
dx)x(fI
Integration
4
The process of measuring
the area under a curve.
Here:
f(x) is the integrand
a= lower limit of integration
b= upper limit of integration
f(x)
a b
y
x
ò
b
a
dx)x(f

5. Basic Idea of Simpson’s 1/3
rd
Rule
5
Divide the X-axis into equally spaced divisions
Apply Simpson’s 1/3rd Rule over each interval
Sum up those result
f(x)
. . .
x
0
x
2
x
n-2
x
n
x
.....dx)x(fdx)x(fdx)x(f
x
x
x
x
b
a
++= òòò
4
2
2
0
òò
-
-
-
++
n
n
n
n
x
x
x
x
dx)x(fdx)x(f....
2
2
4

6. Basis of Simpson’s 1/3
rd
Rule
Simpson’s 1/3rd rule is an extension of Trapezoidal rule
where the integrand is approximated by a second order
polynomial.
6
Hence
òò »=
b
a
b
a
dx)x(fdx)x(fI
2
Where is a second order polynomial. )x(f
2
2
2102 xaxaa)x(f ++=

7. Basis of Simpson’s 1/3
rd
Rule
7
Choose
)),a(f,a( ,
ba
f,
ba
÷
ø
ö
ç
è
æ
÷
ø
ö
ç
è
æ++
22
))b(f,b(and
as the three points of the function to evaluate a
0
, a
1
and a
2
.
2
2102 aaaaa)a(f)a(f ++==
2
2102
2222
÷
ø
ö
ç
è
æ+
+÷
ø
ö
ç
è
æ+
+=÷
ø
ö
ç
è
æ+
=÷
ø
ö
ç
è
æ+ ba
a
ba
aa
ba
f
ba
f
2
2102 babaa)b(f)b(f ++==

8. Basis of Simpson’s 1/3
rd
Rule
Solving the previous equations for a
0
, a
1
and a
2
give
22
22
0
2
2
4
baba
)a(fb)a(abf
ba
abf)b(abf)b(fa
a
+-
++÷
ø
ö
ç
è
æ+
-+
=
221
2
2
433
2
4
baba
)b(bf
ba
bf)a(bf)b(af
ba
af)a(af
a
+-
+÷
ø
ö
ç
è
æ+
-++÷
ø
ö
ç
è
æ+
-
-=
222
2
2
22
baba
)b(f
ba
f)a(f
a
+-
÷
ø
ö
ç
è
æ
+÷
ø
ö
ç
è
æ+
-
=

9. Basis of Simpson’s 1/3
rd
Rule
9
Then
ò»
b
a
dx)x(fI
2
( )ò ++=
b
a
dxxaxaa
2
210
b
a
x
a
x
axa
ú
û
ù
ê
ë
é
++=
32
3
2
2
10
32
33
2
22
10
ab
a
ab
a)ab(a
-
+
-
+-=

10. Basis of Simpson’s 1/3
rd
Rule
10
Substituting values of a
0
, a
1
, a
2
give
ú
û
ù
ê
ë
é
+÷
ø
ö
ç
è
æ+
+
-
=ò )b(f
ba
f)a(f
ab
dx)x(f
b
a 2
4
6
2
Since for Simpson’s 1/3rd Rule, the interval [a, b] is broken
into 2 segments, the segment width
2
ab
h
-
=
it is called Simpson’s 1/3rd Rule.

11. 11
Important points to be considered while applying
Simpson’s 1/3 Rule are:
Number of intervals must be an even
number.
Minimum of 3 points are required
Intervals are expected to be equal

12. THE END
•http://numericalmethods.eng.usf.edu
•http://esurveying.net/land-survey/earthwork-area-calculation-simpsons-13rd-rule
•http://mathworld.wolfram.com/SimpsonsRule.html