This document describes numerical integration and differentiation techniques taught in a B.Tech Engineering Mathematics course. It covers the Trapezoidal, Simpson's 1/3 and 3/8 rules for numerical integration of functions. For numerical differentiation, it discusses Euler's method, Picard's method, and Taylor series for solving ordinary differential equations. Examples are provided to illustrate the application of these numerical methods to evaluate integrals and solve initial value problems.
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we are student of Daffodil International University .
My teammate was Fatema Akter , Rashedul Islam And the respected teacher was Hasin rehana
Lecturer
Faculty of Science and Information Technology
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Role of optimization in engineering design is prominent one with the a dvent of computers. Optimization has become a part of computer aided design methodology. It is primarily being used in those design activities in which the goal is not only to achieve a feasible design,but als o a design objective. The paper reviews the optimization in detail followed by the literature review and b rief discussion of Hooks and Jeeves Method Analysis with an example.
These slides contain information about Euler method,Improved Euler and Runge-kutta's method.How these methods are helpful and applied to our questions are detailed discussed in the slides.
Presentation of the work on Prime Numbers.
intended for mathematics loving people.
Please send comments and suggestions for improvement to solo.hermelin@gmail.com.
More presentations can be found in my website at http://solohermelin.com.
Gaussian Quadrature Formulas, which are simple and will help learners learn about Gauss's One, Two and Three Point Formulas, I have also included sums so that learning can be easy and the method can be understood.
Hi
we are student of Daffodil International University .
My teammate was Fatema Akter , Rashedul Islam And the respected teacher was Hasin rehana
Lecturer
Faculty of Science and Information Technology
Review of Hooke and Jeeves Direct Search Solution Method Analysis Applicable ...ijiert bestjournal
Role of optimization in engineering design is prominent one with the a dvent of computers. Optimization has become a part of computer aided design methodology. It is primarily being used in those design activities in which the goal is not only to achieve a feasible design,but als o a design objective. The paper reviews the optimization in detail followed by the literature review and b rief discussion of Hooks and Jeeves Method Analysis with an example.
These slides contain information about Euler method,Improved Euler and Runge-kutta's method.How these methods are helpful and applied to our questions are detailed discussed in the slides.
Presentation of the work on Prime Numbers.
intended for mathematics loving people.
Please send comments and suggestions for improvement to solo.hermelin@gmail.com.
More presentations can be found in my website at http://solohermelin.com.
Gaussian Quadrature Formulas, which are simple and will help learners learn about Gauss's One, Two and Three Point Formulas, I have also included sums so that learning can be easy and the method can be understood.
Lecture slides introducing Numerical Integration with the Trapezoidal Rule and Simpson's 1/3 Rule. Some parts of this presentation are based on resources at http://nm.MathForCollege.com, primarily http://mathforcollege.com/nm/topics/trapezoidal_rule.html and http://mathforcollege.com/nm/topics/simpsons_13rd_rule.html
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Numerical integration is the approximate computation of an integral using numerical techniques. The numerical computation of an integral is sometimes called quadrature. ... A generalization of the trapezoidal rule is Romberg integration, which can yield accurate results for many fewer function evaluations.
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Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
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2. Unit-V Numerical Integration and Numerical Differentiation
Content
NUMERICAL INTEGRATION—Trapezoidal method & it’s problems method, Simpson’s one
third and three-eight rules & Problem based on Simpson’s one third and three-eight rules.
NUMERICAL DIFFERENTIATION—Solution of ordinary differential equations by following
methods: Euler’s Method, Picard’s Method and forth-order Runge- Kutta methods & it’s
problems
3. Unit-V Numerical Integration and Numerical Differentiation
1.1 Numerical Integration— The process of evaluating a definite integral from a set of
tabulated values of the integrand ( ) is called numerical integration. When this process is
applied to a function of single function is called quadrature.
Consider the definite integral∫ ( ) representsthe area between = ( ) with random
= and = .This integration is possible only when ( ) is explicitly given or otherwise it
is not possible to evaluate.
In numerical integration for given set of ( + 1) paired values of the function taking the
values , , … corresponding to the values , , … , where ( ) is not known
explicitly, it is possible to compute∫ ( ) by numerical integration by using various method.
1. Trapezoidal method
2. Simpson’s one third rule
3. Simpson’s three-eight rule
1.2 Newton-Cote’s quadrature— Let
= ( )
Where ( ) takes the values , , … , for , , … divide the interval ( , ) into n
sub-interval of width h, so that = , = + ℎ, = + 2ℎ, = + ℎ = .
Then = ∫ ( )
Putting = + ℎ
⟹ = ℎ
∴ = ℎ ( + ℎ)
By Newton’s forward interpolation formula
= ℎ + ∆ +
( − 1)
2!
∆ +
( − 1)( − 3)
3!
∆ + ⋯
Integration term by term, we get
4. Unit-V Numerical Integration and Numerical Differentiation
( )
= ℎ +
2
∆ +
(2 − 3)
12
∆ +
( − 2)
24
∆ + ⋯ … … ( )
This equation is known as Newton-Cote’s quadrature formula. Being a general formula, we
deduce many formula’s from this by taking = 1,2,3 …
1.3 Trapezoidal method— Assume that ( ) is continuous on [ , ] and divide [ , ] into n
subinterval of equal length.
∆ =
−
Using ( + 1) points = , = + ∆ , = + 2∆ , = + ∆ =
Computing the values of ( ) at these points
= ( ), = ( ), = ( ), = ( )
Approximate integral by using n trapezoids formed by using straight line segments between two
points ( , ) and ( , ) for 1 ≤ ≤ as shown in the figure:
Area of a trapezoid is obtained by adding the areas of rectangles and triangles.
= ∆ +
1
2
( − )∆ =
( )∆
2
Adding area of the n trapezoids, the approximation is
( ) ≈
( + )∆
2
+
( + )∆
2
+
( + )∆
2
+ ⋯ +
( + )∆
2
5. Unit-V Numerical Integration and Numerical Differentiation
This simplifies the trapezoidal rule.
( ) ≈
∆
2
( + 2 + 2 + ⋯ + 2 + )
≈
∆
[( + ) + 2( + + ⋯ + )]
We can also replace ∆ withℎ. So the formula will be
( ) =
ℎ
2
[( + ) + 2( + + ⋯ + )]
≈ [( ℎ ) + 2( ℎ )]
Another Procedure—
Putting = 1in ( ) and taking the curve through ( , ) and ( , ) as straight line. i.e.
Polynomial of first order so that differences of order higher than first become zero, we get
( ) = ℎ +
1
2
∆ =
ℎ
2
( + )
Similarly
( ) = ℎ +
1
2
∆ =
ℎ
2
( + )
⋮
( ) =
ℎ
2
( )
( + )
Adding these n integrals, we obtain
( ) =
ℎ
2
[( + ) + 2( + + ⋯ + )]
This is known as the trapezoidal rule.
6. Unit-V Numerical Integration and Numerical Differentiation
Example—Evaluate∫ by using Trapezoidal rule. Verify result by actual integration.
Solution—given that ( ) =
Interval length ( – ) = (3 – (−3) ) = 6
So we divide 6 equal intervals with ℎ = 6/6 = 1.0
And tabulate the values as below
-3 -2 -1 0 1 2 3
= 81 16 1 0 1 16 81
We know that—
( ) ≈
ℎ
2
[( + ) + 2( + + ⋯ + )]
≈
1
2
[(81 + 81) + 2 (16 + 1 + 0 + 1 + 16)] =
162 + 68
2
= 115
By actual integration ∫ = − − = + = = 97.5
Example— Evaluate ∫ ( )
by using Trapezoidal rule with h = 0.2.
Solution— Given ( ) = ( )
and interval length ( – ) = (1 – 0 ) = 1.
So we divide 6 equal intervals with h= 0.2
We know ∫ ( ) ≈ [( + ) + 2( + + ⋯ + )]
1
(1 + )
≈
0.2
2
[(1 + 0.5000) + 2(0.96154 + 0.86207 + 0.73529 + 0.60976)]
=(0.1)[ (1.05) + 6.33732 ]
= 0.783732
0 0.2 0.4 0.6 0.8 1
y =
1
(1 + x )
1 0.96154 0.86207 0.73529 0.60976 0.5000
7. Unit-V Numerical Integration and Numerical Differentiation
1.4 Simpson’s one third rule— Putting = 2 in ( ) and taking the curve through
( , ), ( , ) and ( , ) as a parabola, i.e. a polynomial of second order so that differences
of higher than second vanish, we get
( ) = 2ℎ + ∆ +
1
6
∆ =
ℎ
3
( + 4 + )
Similarly
( ) =
ℎ
3
( + 4 + )
⋮
∫ ( ) =( )
( + 4 + ), is even.
Adding these n integrals, we have when is even
( ) =
ℎ
3
[( + ) + 4( + + ⋯ + ) + 2( + + ⋯ + )]
=(ℎ/3) [ (sum of the irst and last ordinates ) + 2 (Sum of remaining even ordinates)
+4 ( sum of remaining odd ordinates) ]
This is known as the Simpson’s one third rule or simply Simpson’s rule.
1.4 Simpson’s three-eight rule — Putting = 3 in ( ) and taking the curve through
( , ): = 0,1,2,3 as polynomial of third order so that the differences above the third order
vanish, we get
( ) = 3ℎ +
3
2
∆ +
3
2
∆ +
1
8
∆ =
3ℎ
8
( + 3 + 3 + )
Similarly, ∫ ( ) = ( + 3 + 3 + ) and so on.
adding all these expressions from to + ℎ, where n is multiple of 3, we obtain
( ) =
3ℎ
8
[( + ) + 3( + + + … + ) + 2( + + ⋯ + )]
= (3ℎ/8) [ (sum of the irst and last ordinates )
+ 2 (Sum of multiples of three ordinates) + 3 ( sum of remaining ordinates)]
Which is known as Simpson’s three-eight rule.
8. Unit-V Numerical Integration and Numerical Differentiation
Example— Evaluate∫ by using Simpson’s one third rule and Simpson’s three-eight
rule. Verify result by actual integration.
Solution—We are given that ( ) =
Interval length ( – ) = (3 – (−3) ) = 6
So we divide 6 equal intervals with ℎ = 6/6 = 1.0 and tabulate the values as below
-3 -2 -1 0 1 2 3
= 81 16 1 0 1 16 81
By Simpson’s one third rule
=
ℎ
3
[( + ) + 4( + + ) + 2( + )]
= [(81 + 81) + 4(16 + 0 + 16) + 2(1 + 1)] = = 98
By Simpson’s three-eight rule
=
3ℎ
8
[( + ) + 3( + + + ) + 2 )]
= [(81 + 81) + 3(16 + 1 + 1 + 16) + 2 × 0] = = 99
By actual integration ∫ x dx = − − = + = = 97.5
Example—Evaluate∫
.
by using Trapezoidal rule, Simpson’s one third rule and
Simpson’s three-eighth rule.
Solution—We are given that ( ) = Interval length ( – ) = (5.2 – 4 ) = 1.2.So
we divide 6 equal intervals with ℎ = 0.2 and tabulate the values as below
4.0 4.2 4.4 4.6 4.8 5.0 5.2
= 1.39 1.44 1.48 1.53 1.57 1.61 1.65
By Trapezoidal rule
=
.
ℎ
2
[( + ) + 2( + + ⋯ + )]
=
0.2
2
[(1.39 + 1.65) + 2 (1.44 + 1.48 + 1.53 + 1.57 + 1.61)]
= (0.1) [ 3.04 + 2(7.63) ]
= 1.83
9. Unit-V Numerical Integration and Numerical Differentiation
By Simpson’s one third rule
.
=
ℎ
3
[( + ) + 4( + + ) + 2( + )]
= (0.2/3) [ (1.39 + 1.65) + 2 (1.48 + 1.57) + 4 (1.44 + 1.53 + +1.61) ]
= (0.0667) [ 3.04 + 2(3.05) + 4 (4.58) ]
= 1.83
By Simpson’s three-eight rule
.
=
3ℎ
8
[( + ) + 3( + + + ) + 2 )]
= (
× .
)[ (1.39 + 1.65) + 2 (1.53) + 3 (1.44 + 1.48 + 1.57 + +1.61) ]
= (0.075 ) [ 3.04 + 3.06 + 3 (6.1) ]
= 1.83
2 Numerical Differentiations—
2.1 Picard’s Method— Let us considers the first order differential equation = ( , ) and
( ) = then from the Picard’s method, nth
approximation to the solution of Initial value
problem eq (i) is
= +∫ ( , )
Example —Use Picard’s method to solve = − upto the fourth approximation, when
(0) = 1.
Solution—Given differential equation is = − , when (0) = 1.
Picard’s formula says
= +∫ ( , )
= − = 1 − × 1 = 1 −
2
= − = 1 − 1 −
2
= 1 − −
2
= 1 −
2
+
8
= − = 1 − 1 −
2
+
8
= 1 − −
2
+
8
= 1 −
2
+
8
−
48
16. Unit-V Numerical Integration and Numerical Differentiation
Exercise
1. If = 2 − and = 2when = 1, perform three iterations of Picard’s method to estimate a value
for y when x= 1.2. Work to four places of decimals throughout and state how accurate is the result of the
third iteration.
2. Solve ′ = − ,and y(0) = 1, determine the values of y at x =(0.01)(0.01)(0.04) by Euler’s method.
3. Obtain the values of y at x= 0.1, 0.2 using R.K. method of fourth order for the differential equation ′
=
− , given y(0) =1.5.
4. Apply Rungakutta method to find an approximation value of , when = 0 given that = − , =
0,when = 0 with ℎ = 0.1
5. Find y (0.2) given = – , (0) = 2 taking h = 0.1. by Runge –Kutta method.
6. Evaluate y(1.4) given = + , (1.2) = 2. By Runge-Kutta Method.
17. Unit-V Numerical Integration and Numerical Differentiation
Reference
1. http://en.wikipedia.org/wiki/File:Integral_as_region_under_curve.svg
2. http://www.google.co.in/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0CAcQ
jRw&url=http%3A%2F%2Fcalculator.mathcaptain.com%2Ftrapezoidal-rule-
calculator.html&ei=z7qnVKa5DsiNuATbtoHoCQ&bvm=bv.82001339,d.c2E&psig=AFQjCNHqSN98kFMHM
wdx482zLm5KtRJA6A&ust=1420364847649006
3. Numerical Method for Science and Computer science by M.K. Jain, S.R.K. Iyenger, R.K. Jain
4. Higher Engineering Mathematics, B.S. Grewal, Khanna Publishers.
5. Higher Engineering Mathematics, B V Ramana, McGraw Hill Education