The document discusses numerical integration methods including Boole's rule, Weddle's rule, Gaussian quadrature, and Newton-Cotes formulas. Specific examples illustrate the application and accuracy of these methods for estimating definite integrals. The document highlights practical applications and error calculation techniques related to these numerical methods.

2. Group
Members
2
Rahima Khalid
G1F18BSCS0009
Samavia Ameen
G1F18BSCS0146
Hadia
G1F18BSCS0124

3. Boole’s Rule
and
Weddle’s Rule
3
Rahima Khalid
G1F18BSCS0009

4. Boole’s
Rule
Boole's rule, named after
George Boole, is a method of
numerical integration.
4

5. Boole’s
Rule
Example:
Use the Boole’s Rule with n=8 to
estimate
1
5
1 + 𝑥2 ⅆ𝑥
5

6. SOLUTION:
6
For n= 8, we have Δ𝑥 = ℎ =
5−1
8
= 0.5 . We compute the
values of 𝑦0, 𝑦1, 𝑦2,…..𝑦n.
x 1 1.5 2 2.5 3 3.5 4 4.5 5
1 + 𝑥2 2 3.25 5 7.25 10 13.25 17 21.25 26

7. 7
SINCE THE FORMULA IS:
•
𝛼
𝛽
𝑦𝑑𝑥 ≈
2ℎ
45
[ 7(𝑦0+ 𝑦n) + 32(𝑦1+ 𝑦3+ 𝑦5 +…+ 𝑦n−3+
𝑦n−1 )+ 12(𝑦2+ 𝑦6+𝑦10 +…+ 𝑦n−2) + 14 (𝑦4+ 𝑦8+…
+𝑦n−4)
REDUCING THE FORMULA TO N=8:

𝛼
𝛽
𝑦𝑑𝑥 ≈
2ℎ
45
[ 7(𝑦0+ 𝑦n) + 32(𝑦1+ 𝑦3+ 𝑦5 + 𝑦7 )+
12(𝑦2+ 𝑦6) + 14 (𝑦4)

8. 8
1
5
1 + 𝑥2𝑑𝑥 ≈
2(0.5)
45
[ 7( 2+ 26) +
32( 3.25+ 7.25+ 13.25+ 21.25)+ 12( 5+ 17)
+ 14 ( 10) ≈ 12.75601175
Exact Solution is:
1
5
1 + 𝑥2𝑑𝑥 ≈ 12.75597
Difference= 12.75601175 - 12.75597 = 0.0000417

9. “
9
Weddles’s
Rule

10. 10
Example:
Use the Weddle’s Rule with n=6 to estimate
0
𝜋
sin 𝑥 ⅆ𝑥
(Mechanical Quadrature)
Weddle's rule is a method of integration used for computing
definite integral.

11. Solution
11
For n= 6, we have Δ𝑥 = ℎ =
𝜋
6
.
We compute the values of 𝑦0,
𝑦1, 𝑦2,…..𝑦6.
0 1 2 4 5 5 6
x 0 𝜋
6
2𝜋
6
3𝜋
6
4𝜋
6
5𝜋
6
6𝜋
6
sin 𝑥 sin 0 sin
𝜋
6 sin
2𝜋
6
sin
3𝜋
6
sin
4𝜋
6
sin
5𝜋
6
sin
6𝜋
6
sin 𝑥 0 1
2
3
2
1 3
2
1
2
0

12. Solution
12
SINCE THE FORMULA IS:
𝛼
𝛽
𝑦𝑑𝑥 ≈
3ℎ
10
[𝑦0+5 𝑦1+ 𝑦2 + 6 𝑦3+ 𝑦4 + 5𝑦5+ 𝑦6]
Therefore,
0
𝜋
𝑠𝑖𝑛𝑥𝑑𝑥 ≈
3(
𝜋
6
)
10
[0+5
1
2
+
3
2
+6(1)+
3
2
+ 5
1
2
+0] ≈
1.999994586
Exact Solution is:
0
𝜋
𝑠𝑖𝑛𝑥𝑑𝑥 = 2
Difference= 2-1.999994586 = 0.000005414

13. Quadrature
Method
13
Samavia Ameen
G1F18BSCS0146

14. Introduction
14
Numerical Integration
 Newton cotes formula
(Nodes are evenly spread)
 Guassian Quadrature
(Nodes are chosen)

15. The term Quadrature refers to methods in
which the points where the function is
evaluated are chosen.
So that the formula is exact for polynomials of
as high degree as possible.
The most basic of these methods is Gaussian
Quadrature.

16. 16
The general form of Gaussian quadrature is
−1
1
𝑓 𝑥 𝑑𝑥 ≈
𝑖=1
𝑛
𝑐𝑖 𝑓 𝑥𝑖
𝑐𝑖 and 𝑥𝑖 depends upon choice of n
The integration formula is exact for polynomials of degree 2n-1
−1
1
𝑓 𝑥 𝑑𝑥 ≈ 𝑐1 𝑓 𝑥1 +𝑐2 𝑓 𝑥2 +𝑐3 𝑓 𝑥3 +-----------𝑐𝑛 𝑓 𝑥𝑛

17. 17
For n=1 𝑥0, 𝑥1
−1
1
𝑓 𝑥 𝑑𝑥 ≈ 𝑐1 𝑓 𝑥1 is exact for polynomials of
degree upto 2.1-1=1
𝑥0
= −1
1
1𝑑𝑥 = ǀ𝑥 ǀ−1
1
= 1-(-1)= 2
RHS =𝑐1.1 𝑐1=2
𝑥1= −1
1
𝑥𝑑𝑥 = ǀ
𝑥2
2
ǀ−1
1
=
1
2
−
1
2
=0
RHS =𝑐1. 𝑥1= 2𝑥1 0 = 2𝑥1 𝑥1 = 0
−𝟏
𝟏
𝒇 𝒙 𝒅𝒙= 2𝒇 𝒙

18. 18
For n=2 𝑥0
, 𝑥1
, 𝑥2
, 𝑥3
−1
1
𝑓 𝑥 𝑑𝑥 ≈ 𝑐1𝑓 𝑥1 + 𝑐2𝑓 𝑥2 upto degree
2.2-1=3
𝑥0= −1
1
1𝑑𝑥=2 =𝑐1 + 𝑐2 𝑐1 + 𝑐2=2 …i
𝑥1= −1
1
𝑥𝑑𝑥=0 =𝑐1𝑥1 + 𝑐2𝑥2 𝑐1𝑥1 + 𝑐2𝑥2=0 ....ii
𝑥2= −1
1
𝑥2𝑑𝑥 =
2
3
=𝑐1𝑥1
2
+ 𝑐2𝑥2
2
𝑐1𝑥1
2
+ 𝑐2𝑥2
2
=
2
3
...iii
𝑥3
= −1
1
𝑥3
𝑑𝑥=0 =𝑐1𝑥1
3
+ 𝑐2𝑥2
3
𝑐1𝑥1
3
+ 𝑐2𝑥2
3
=0 ...iv
−𝟏
𝟏
𝒇 𝒙 𝒅𝒙= 𝒇
−1
3
+ 𝒇
1
3

19. 19
For n=3 2.3-1=5
−𝟏
𝟏
𝒇 𝒙 𝒅𝒙=
5
9
𝒇 −
𝟑
𝟓
+
8
9
𝒇 𝟎 +
5
9
𝒇(
𝟑
𝟓
)
For n=2
−𝟏
𝟏
𝒇 𝒙 𝒅𝒙= 𝒇
−1
3
+ 𝒇
1
3
For n=1
−𝟏
𝟏
𝒇 𝒙 𝒅𝒙= 2𝒇 𝒙

20. Example
20
−𝟏
𝟏
𝒆−𝒙𝟐
𝒅𝒙 =?
For n=1
−1
1
𝑒−𝑥2
𝑑𝑥 = 2𝑓 0 = 2𝑒−02
=2
For n=2
−1
1
𝑒−𝑥2
𝑑𝑥 = 𝒇
−1
3
+ 𝒇
1
3
=𝑒
−1
3 + 𝑒
−1
3
=2𝑒
−1
3 ≈2(0.7165) ≈1.433
For n=3
−1
1
𝑒−𝑥2
𝑑𝑥 =
5
9
𝒇 −
𝟑
𝟓
+
8
9
𝒇 𝟎 +
5
9
𝒇(
𝟑
𝟓
)
=
5
9
𝑒
−3
5 +
8
9
. 𝟏 +
5
9
𝑒
−3
5 =
10
9
𝑒
−3
5 +
8
9
≈1.49868

21. Newton Cote’s
Formula
21
Hadia
G1F18BSCS0124

22. Newton
Cotes’s
Formula
 Also known as Newton–Cotes quadrature rules.
 A group of formulas for numerical integration
(quadrature) based on evaluating the integrand
at equally spaced points
 They are named after Isaac Newton and Roger
Cotes.
22

23. Newton
Cotes’s
Formula
Useful if the value of the integrand at equally
spaced points is given.
 If it is possible to change the points at
which the integrand is evaluated, then other
methods such as Gaussian quadrature and
Clenshaw–Curtis quadrature are probably
more suitable.
23

24. 24
 It is assumed that value of the function f defined
on[a,b] is known at equally spaced point x,for i=0,.....n
where 𝑥0=a and 𝑥1=b.
 Solved using Newton Cortes Formula
 There are two types of Newton Cortes Formula.
1. The Closed Type
2. The Open Type

25. The
Closed
Newton
Cotes’s
Formula
n this type we uses the function value at all
values.
The closed newton cotes formula of degree n
is stated as
𝑎
𝑏
𝑓𝑥 𝑑𝑥 =
𝑖=0
𝑛
𝑤𝑖. 𝑓(𝑥𝑖)
where 𝑥𝑖=ℎ𝑖+𝑥0,with h equal to
𝑥𝑛−𝑥0
𝑛
=
(𝑏−𝑎)
𝑛
.
The 𝑤𝑖 are called weights.
25

26. 26
 The Closed Newton Cortes Formula produced
different methods while dealing with different degrees.

27. The
Opened
Newton
Cotes’s
Formula
In this type we cant used the functions value
at end points.
The opened newton cotes formula of degree n
is stated as
𝑎
𝑏
𝑓𝑥 𝑑𝑥 =
𝑖=1
𝑛−1
𝑤𝑖. 𝑓(𝑥𝑖)
The weights are found in manner simliar to
closed formula.
27

28. 28
 The Closed Newton Cortes Formula produced
different methods while dealing with different degrees.

29. Practical
Life
Example
29
The vertical distance is covered by a rocket
from t=8 to t=30 seconds is given by
x =
8
30
2000ln
140000
140000 − 2100t
− 9.8t ⅆt
(a)use single segment trapezoid rule to find
the DISTANCE COVERED?
(b)Find the True Error?
(c)Find the Absolute Realtive True Error?

30. Practical
Life
Example
30
(a) 𝜤 ≈ 𝒃 −
𝒇(𝒂)+𝒇(𝒃)
𝟐
a=8,b=30
𝑓(𝑡) = 2000ln
140000
140000 − 2100t
− 9.8t
𝑓(𝑡) = 2000ln
140000
140000 − 2100(8)
− 9.8(8)
= 177.27m/s
𝑓(𝑡)
= 2000ln
140000
140000 − 2100(30)
− 9.8(30)

31. Practical
Life
Example
31
(a) 𝜤=(30-8)
177.27+001.67
2
= 11868𝑚
(b)The Exact value of the above integral is
x =
8
30
2000ln
140000
140000 − 2100t
− 9.8t ⅆt = 11061m
𝐸𝑡=True Value - Approximate Value
=11061-11868
= -807m
(c)The Absolute Realtive Ture Error would be
𝛦𝑡 =
11061 − 11868
11061
× 100
=7.2959%

32. 32