Numerical Integration (Gauss-
Legendre Two Point Formula)
Compiled by
Dr. Shyam Arjun Sonawane
Associate Professor, Mechanical Engineering
Government College of Engineering & Research, Avasari (Kh)
What is Integration ?
• The process of measuring the area under a function plotted on a
graph.
• Where:
• f(x) is the integrand
• a= lower limit of integration
• b= upper limit of integration
f(x)
a
b
b
a
dx)x(f
y
x
=
b
a
dx)x(fI
3
Limitations of Newton-Cotes Formula
• When we calculate integration by Newton-Cotes method, the area is divided
in number of strips with step size h.
• The formula includes y values (function values) at the corresponding x
values.
• But for complicated equations it becomes difficult to find function values.
• As more processing time is required, computer program also becomes
inefficient.
• If we reduce the number of strips, number of function values will be less.
But, this affects the accuracy of the solution.
• This limitation of Newton-Cotes equation is overcome by Gauss-Legendre
quadrature method.
Gauss-Legendre Quadrature Method
This method converts the function variable f(x) to a function variable
f(u) such that the original limits x
0
and x
n
gets changed to (-1) and
(+1) respectively whereas the area under the curve f(x) and f(u)
remains same. Refer figure (a) and (b)
Gauss-Legendre Quadrature Method …..Contd
Gauss-Legendre Quadrature Method …..Contd
Gauss-Legendre Two Point Formula
Gauss-Legendre Two Point Formula …..Contd
Gauss-Legendre Two Point Formula …..Contd
Gauss-Legendre Two Point Formula …..Contd
Gauss-Legendre Two Point Formula …..Contd
Gauss-Legendre Two Point Formula …..Contd
Problems on Gauss-Legendre two point
formula
Problems on Gauss-Legendre two point
formula …..Contd
Reference Books
1. Steven C. Chapra, Raymond P. Canale, Numerical Methods for
Engineers, 4/e, Tata McGraw Hill Editions
2. Dr. B. S. Garewal, Numerical Methods in Engineering and Science,
Khanna Publishers,.
3. Steven C. Chapra, Applied Numerical Methods with MATLAB for
Engineers and Scientist, Tata Mc-Graw Hill Publishing Co-Ltd
4. Rao V. Dukkipati, Applied Numerical Methods using Matlab, New
Age International Publishers
Reference Books …Contd
5. Gerald and Wheatley, Applied Numerical Analysis, Pearson
Education Asia
6. E. Balagurusamy, Numerical Methods, Tata McGraw Hill
7. P. Thangaraj, Computer Oriented Numerical Methods, PHI
8. S. S. Sastry, Introductory Methods of Numerical Analysis, PHI.
Numerical Integration (Gauss-
Legendre Three Point Formula)
Compiled by
Dr. Shyam Arjun Sonawane
Associate Professor, Mechanical Engineering
Government College of Engineering & Research, Avasari (Kh)
What is Integration ?
• The process of measuring the area under a function plotted on a
graph.
• Where:
• f(x) is the integrand
• a= lower limit of integration
• b= upper limit of integration
f(x)
a
b
b
a
dx)x(f
y
x
=
b
a
dx)x(fI
3
Limitations of Newton-Cotes Formula
• When we calculate integration by Newton-Cotes method, the area is divided
in number of strips with step size h.
• The formula includes y values (function values) at the corresponding x
values.
• But for complicated equations it becomes difficult to find function values.
• As more processing time is required, computer program also becomes
inefficient.
• If we reduce the number of strips, number of function values will be less.
But, this affects the accuracy of the solution.
• This limitation of Newton-Cotes equation is overcome by Gauss-Legendre
quadrature method.
Gauss-Legendre Quadrature Method
This method converts the function variable f(x) to a function variable
f(u) such that the original limits x
0
and x
n
gets changed to (-1) and
(+1) respectively whereas the area under the curve f(x) and f(u)
remains same. Refer figure (a) and (b)
Gauss-Legendre Quadrature Method …..Contd
Gauss-Legendre Quadrature Method …..Contd
Gauss-Legendre Three Point Formula
Gauss-Legendre Three Point Formula ..Contd
Gauss-Legendre Three Point Formula ..Contd
Gauss-Legendre Three Point Formula ..Contd
Gauss-Legendre Three Point Formula ..Contd
Gauss-Legendre Three Point Formula ..Contd
Gauss-Legendre Three Point Formula ..Contd
Problems on Gauss-Legendre Three Point
formula
Problems on Gauss-Legendre Three Point
formula …..Contd
Reference Books
1. Steven C. Chapra, Raymond P. Canale, Numerical Methods for
Engineers, 4/e, Tata McGraw Hill Editions
2. Dr. B. S. Garewal, Numerical Methods in Engineering and Science,
Khanna Publishers,.
3. Steven C. Chapra, Applied Numerical Methods with MATLAB for
Engineers and Scientist, Tata Mc-Graw Hill Publishing Co-Ltd
4. Rao V. Dukkipati, Applied Numerical Methods using Matlab, New
Age International Publishers
Reference Books …Contd
5. Gerald and Wheatley, Applied Numerical Analysis, Pearson
Education Asia
6. E. Balagurusamy, Numerical Methods, Tata McGraw Hill
7. P. Thangaraj, Computer Oriented Numerical Methods, PHI
8. S. S. Sastry, Introductory Methods of Numerical Analysis, PHI.
Numerical Double Integration
(Simpson’s 1/3
rd
Rule)
Compiled by
Dr. Shyam Arjun Sonawane
Associate Professor, Mechanical Engineering
Government College of Engineering & Research, Avasari (Kh)
Double Integration
Simpson’s 1/3
rd
Rule
Simpson’s 1/3
rd
Rule …Contd
Simpson’s 1/3
rd
Rule …Contd
Simpson’s 1/3
rd
Rule …Contd
Simpson’s 1/3
rd
Rule …Contd
Simpson’s 1/3
rd
Rule …Contd
Simpson’s 1/3
rd
Rule …Contd
Simpson’s 1/3
rd
Rule …Contd
Simpson’s 1/3
rd
Rule …Contd
Simpson’s 1/3
rd
Rule …Contd
Step 3: Make a table for all values of f(x,y)
.
Simpson’s 1/3
rd
Rule …Contd
Simpson’s 1/3
rd
Rule …Contd
Simpson’s 1/3
rd
Rule …Contd
Simpson’s 1/3
rd
Rule …Contd
Simpson’s 1/3
rd
Rule …Contd
Simpson’s 1/3
rd
Rule …Contd
Problems on Simpson’s 1/3
rd
Rule
Reference Books
1. Steven C. Chapra, Raymond P. Canale, Numerical Methods for
Engineers, 4/e, Tata McGraw Hill Editions
2. Dr. B. S. Garewal, Numerical Methods in Engineering and Science,
Khanna Publishers,.
3. Steven C. Chapra, Applied Numerical Methods with MATLAB for
Engineers and Scientist, Tata Mc-Graw Hill Publishing Co-Ltd
4. Rao V. Dukkipati, Applied Numerical Methods using Matlab, New
Age International Publishers
Reference Books …Contd
5. Gerald and Wheatley, Applied Numerical Analysis, Pearson
Education Asia
6. E. Balagurusamy, Numerical Methods, Tata McGraw Hill
7. P. Thangaraj, Computer Oriented Numerical Methods, PHI
8. S. S. Sastry, Introductory Methods of Numerical Analysis, PHI.
Numerical Integration
(Simpson’s 1/3
rd
Rule)
Compiled by
Dr. Shyam Arjun Sonawane
Associate Professor, Mechanical Engineering
Government College of Engineering & Research, Avasari (Kh)
What is Integration ?
• The process of measuring the area under a function plotted on a
graph.
• Where:
• f(x) is the integrand
• a= lower limit of integration
• b= upper limit of integration
f(x)
a
b
b
a
dx)x(f
y
x
=
b
a
dx)x(fI
3
Newton-Cotes Quadrature Formula
Simpson’s 1/3
rd
Rule
Simpson’s 1/3
rd
Rule …Contd
Simpson’s 1/3
rd
Rule …Contd
Simpson’s 1/3
rd
Rule …Contd
Simpson’s 1/3
rd
Rule …Contd
Simpson’s 1/3
rd
Rule …Contd
Simpson’s 1/3
rd
Rule …Contd
Problems on Simpson’s 1/3
rd
Rule
Time(min) 0 2 4 6 8 10 12
Velocity (km/
hr
)
0 22 30 27 18 7 0
Time(min) 2 4 6 8 10 12 14 16 18 20
V (km/hr) 10 18 25 29 32 20 11 5 2 0
Problems on Simpson’s 1/3
rd
Rule …Contd
x (m) 0 0.25 0.50 0.75 1.00
r (m) 1.00 0.9896 0.9589 0.9089 0.8415
t (hr) 0 1 2 3 4 6 8 11 14
q (N/cm
2
hr
0.05
1.72
5.23 6.38 7.86 8.05
8.03
5.82
0.24
Problems on Simpson’s 1/3
rd
Rule …Contd
x 1 1.1 1.2 1.4 1.6 1.9 2.2
3 4.2 5.8 10 14.5 25.1 40
Problems on Simpson’s 1/3
rd
Rule …Contd
Reference Books
1. Steven C. Chapra, Raymond P. Canale, Numerical Methods for
Engineers, 4/e, Tata McGraw Hill Editions
2. Dr. B. S. Garewal, Numerical Methods in Engineering and Science,
Khanna Publishers,.
3. Steven C. Chapra, Applied Numerical Methods with MATLAB for
Engineers and Scientist, Tata Mc-Graw Hill Publishing Co-Ltd
4. Rao V. Dukkipati, Applied Numerical Methods using Matlab, New
Age International Publishers
Reference Books …Contd
5. Gerald and Wheatley, Applied Numerical Analysis, Pearson
Education Asia
6. E. Balagurusamy, Numerical Methods, Tata McGraw Hill
7. P. Thangaraj, Computer Oriented Numerical Methods, PHI
8. S. S. Sastry, Introductory Methods of Numerical Analysis, PHI.
Numerical Integration
(Simpson’s 3/8
th
Rule)
Compiled by
Dr. Shyam Arjun Sonawane
Associate Professor, Mechanical Engineering
Government College of Engineering & Research, Avasari (Kh)
What is Integration ?
• The process of measuring the area under a function plotted on a
graph.
• Where:
• f(x) is the integrand
• a= lower limit of integration
• b= upper limit of integration
f(x)
a
b
b
a
dx)x(f
y
x
=
b
a
dx)x(fI
3
Newton-Cotes Quadrature Formula
Simpson’s 3/8
th
Rule
Simpson’s 3/8
th
Rule …Contd
Simpson’s 3/8
th
Rule …Contd
Simpson’s 3/8
th
Rule …Contd
Simpson’s 3/8
th
Rule …Contd
Simpson’s 3/8
th
Rule …Contd
Simpson’s 3/8
th
Rule …Contd
Problems on Simpson’s 3/8
th
Rule
Problems on Simpson’s 3/8
th
Rule …Contd
x 0 2.5 5.0 7.5 10.0 12.5
15.0
D 5 5.5 6.0 6.75 6.25 5.5 4.0
t 1 2 3 4 5 6 7
T 81 75 80 83 78 70 60
Problems on Simpson’s 3/8
th
Rule …Contd
8. Using the following data calculate the work done by stretching the
spring that has a spring constant of K=300 N/m from x=0 to x=0.3
m. Use Simpson’s 1/3
rd
and 3/8
th
rule.
F(10
3
N)
0 0.01 0.028 0.046 0.063 0.082 0.11
x, m 0 0.05 0.10 0.15 0.20 0.25 0.30
Reference Books
1. Steven C. Chapra, Raymond P. Canale, Numerical Methods for
Engineers, 4/e, Tata McGraw Hill Editions
2. Dr. B. S. Garewal, Numerical Methods in Engineering and Science,
Khanna Publishers,.
3. Steven C. Chapra, Applied Numerical Methods with MATLAB for
Engineers and Scientist, Tata Mc-Graw Hill Publishing Co-Ltd
4. Rao V. Dukkipati, Applied Numerical Methods using Matlab, New
Age International Publishers
Reference Books …Contd
5. Gerald and Wheatley, Applied Numerical Analysis, Pearson
Education Asia
6. E. Balagurusamy, Numerical Methods, Tata McGraw Hill
7. P. Thangaraj, Computer Oriented Numerical Methods, PHI
8. S. S. Sastry, Introductory Methods of Numerical Analysis, PHI.
Numerical Double Integration
(Trapezoidal Rule)
Compiled by
Dr. Shyam Arjun Sonawane
Associate Professor, Mechanical Engineering
Government College of Engineering & Research, Avasari (Kh)
Double Integration
Trapezoidal Rule
Trapezoidal Rule …Contd
Trapezoidal Rule …Contd
Trapezoidal Rule …Contd
Alternate Method:- Prepare a table for n number of strips as
follows:
Trapezoidal Rule …Contd
Trapezoidal Rule …Contd
Trapezoidal Rule …Contd
Trapezoidal Rule …Contd
Trapezoidal Rule …Contd
Trapezoidal Rule …Contd
Trapezoidal Rule …Contd
Problems on Trapezoidal Rule
Reference Books
1. Steven C. Chapra, Raymond P. Canale, Numerical Methods for
Engineers, 4/e, Tata McGraw Hill Editions
2. Dr. B. S. Garewal, Numerical Methods in Engineering and Science,
Khanna Publishers,.
3. Steven C. Chapra, Applied Numerical Methods with MATLAB for
Engineers and Scientist, Tata Mc-Graw Hill Publishing Co-Ltd
4. Rao V. Dukkipati, Applied Numerical Methods using Matlab, New
Age International Publishers
Reference Books …Contd
5. Gerald and Wheatley, Applied Numerical Analysis, Pearson
Education Asia
6. E. Balagurusamy, Numerical Methods, Tata McGraw Hill
7. P. Thangaraj, Computer Oriented Numerical Methods, PHI
8. S. S. Sastry, Introductory Methods of Numerical Analysis, PHI.
Numerical Integration
(Trapezoidal Rule)
Compiled by
Dr. Shyam Arjun Sonawane
Associate Professor, Mechanical Engineering
Government College of Engineering & Research, Avasari (Kh)
What is Integration ?
• The process of measuring the area under a function plotted on a
graph.
• Where:
• f(x) is the integrand
• a= lower limit of integration
• b= upper limit of integration
f(x)
a
b
b
a
dx)x(f
y
x
=
b
a
dx)x(fI
3
Newton-Cotes Quadrature Formula
Trapezoidal Rule
Trapezoidal Rule …Contd
Trapezoidal Rule …Contd
Trapezoidal Rule …Contd
Trapezoidal Rule …Contd
Trapezoidal Rule …Contd
Problems on Trapezoidal Rule
Problems on Trapezoidal Rule …..Contd
x 1 1.1 1.2 1.4 1.6 1.9 2.2
F(x) 3.123 4.247 5.635 9.299 14.303 24.759 39.319
x 1 1.2 1.4 1.6 1.8 2
y 2 2.2 2.7 2.8 3 2.6
Problems on Trapezoidal Rule …..Contd
x, m
0 2 3 4 6 8 10
P g/cm
2
4.00 3.95 3.89 3.80 3.60 3.41 3.30
100 103 106 110 120 133 150
Problems on Trapezoidal Rule …..Contd
Reference Books
1. Steven C. Chapra, Raymond P. Canale, Numerical Methods for
Engineers, 4/e, Tata McGraw Hill Editions
2. Dr. B. S. Garewal, Numerical Methods in Engineering and Science,
Khanna Publishers,.
3. Steven C. Chapra, Applied Numerical Methods with MATLAB for
Engineers and Scientist, Tata Mc-Graw Hill Publishing Co-Ltd
4. Rao V. Dukkipati, Applied Numerical Methods using Matlab, New
Age International Publishers
Reference Books …Contd
5. Gerald and Wheatley, Applied Numerical Analysis, Pearson
Education Asia
6. E. Balagurusamy, Numerical Methods, Tata McGraw Hill
7. P. Thangaraj, Computer Oriented Numerical Methods, PHI
8. S. S. Sastry, Introductory Methods of Numerical Analysis, PHI.
