Derivation of Simpson's 1/3 rule

1. Hello!
1

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

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

4. What is Integration?
∫=
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
∫
b
a
dx)x(f

5. Basic Idea of Simpson’s 1/3rd
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)
. . .
x0 x2 xn-2 xn
x
.....dx)x(fdx)x(fdx)x(f
x
x
x
x
b
a
++= ∫∫∫
4
2
2
0
∫∫
−
−
−
++
n
n
n
n
x
x
x
x
dx)x(fdx)x(f....
2
2
4

6. Basis of Simpson’s 1/3rd
Rule
Simpson’s 1/3rd rule is an extension of Trapezoidal rule
where the integrand is approximated by a second order
polynomial.
6
Hence
∫∫ ≈=
b
a
b
a
dx)x(fdx)x(fI 2
Where is a second order polynomial.)x(f2
2
2102 xaxaa)x(f ++=

7. Basis of Simpson’s 1/3rd
Rule
7
Choose
)),a(f,a( ,
ba
f,
ba











 ++
22
))b(f,b(and
as the three points of the function to evaluate a0, a1 and a2.
2
2102 aaaaa)a(f)a(f ++==
2
2102
2222





 +
+




 +
+=




 +
=




 + ba
a
ba
aa
ba
f
ba
f
2
2102 babaa)b(f)b(f ++==

8. Basis of Simpson’s 1/3rd
Rule
Solving the previous equations for a0, a1 and a2 give
22
22
0
2
2
4
baba
)a(fb)a(abf
ba
abf)b(abf)b(fa
a
+−
++




 +
−+
=
221
2
2
433
2
4
baba
)b(bf
ba
bf)a(bf)b(af
ba
af)a(af
a
+−
+




 +
−++




 +
−
−=
222
2
2
22
baba
)b(f
ba
f)a(f
a
+−






+




 +
−
=

9. Basis of Simpson’s 1/3rd
Rule
9
Then
∫≈
b
a
dx)x(fI 2
( )∫ ++=
b
a
dxxaxaa 2
210
b
a
x
a
x
axa 





++=
32
3
2
2
10
32
33
2
22
10
ab
a
ab
a)ab(a
−
+
−
+−=

10. Basis of Simpson’s 1/3rd
Rule
10
Substituting values of a0, a1, a2 give




+




 +
+
−
=∫ )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