Numerical Integration_Misal Gandhi

8/18/2019 Numerical Integration_Misal Gandhi

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NUMERICAL INTEGRATION

Submitted By: Misal GandhiEnrollment No.- 150490728006Branch : M.E. (Production)Subject : Computational Method (2710002)


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Content

• Introduction• Trapezoidal Rule & Example• Simpson's Rule & Example

• Simpson's Rule & Example

• Application of Simpson’s Rule

1

3

38


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INTRODUCTION

• Numerical Integration is a process to find thevalue of area under curve using a fitted orinterpolated polynomial on the basis of dataof the function corresponding to different xvalues.

• This set of data may also be obtained from anyexperimental observation.


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• The process of Numerical Integration is of helpunder following situation,

• 1. When variation of function f(x) is known fromits value of discrete values of argument x but notas closed from expression. If the integration of

such function is needed then numericalintegration comes to help.

• 2. Numerical Integration is also useful in finding

integration of a function that has complicatedform and for its integral value there is difficulty infinding a closed form expression.


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• In situation 2 the function values are firstcalculated at discrete x and then value of integral is calculated by any method of Numerical Integration.

•

Formula for obtaining Numerical Integrationare known as Quadratures.

•

Trapezoidal Rule, Simpson's Rule, Simpson'sRule are various quadratures which make useof different numbers of equally spaced pointsto find the Integration of fitted polynomial.


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NUMERICAL INTEGRATION


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TRAPEZOIDAL RULE

Similarly for next interval [x 1, x2], we get

Similarly for next interval [x n-1, xn], we get

Put n=1 in General equation


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• Combining all these expression, we get

which is known as Trapezoidal Rule.


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Example:Calculate the value of IntegralAns:Taking h = 0.2

By Trapezoidal Rule, we have

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4

log xdx

X 4 4.2 4.4 4.6 4.8 5 5.2

log x 1.38629 1.43508 1.48160 1.52605 1.56861 1.60943 1.64865

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0 1 2 3 4 5 64

log [ 2( )2h

xdx y y y y y y y

0.2[1.38629 2(1.43508 1.48160 1.52605 1.56861 1.60943) 1.64865)2

0.1(18.27648)

1.827648


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SIMPSON'S RULE

Put n=2 in General equation


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Example:The Velocity v of a particle at distance s from point

on its path is given by the following table: find thetime taken to travel 60 meter.

Ans: w.k.that, Velocity so

Integrating

s (meter) 0 10 20 30 40 50 60

v (meter/sec) 47 58 64 65 61 52 38

dsv

dt ds

dt v

60

0

1t dsv


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s 0 10 20 30 40 50 60

0.02127 0.01724 0.01562 0.01538 0.01639 0.01923 0.026391v

By Simpson’s Rule

So, Time taken to travel 60 meter is 1.0633 sec.

13

60

0 6 2 4 1 3 50

1[( ) 2( ) 4( )

3h

ds y y y y y y yv

60

0 6 2 4 1 3 50

1[( ) 2( ) 4( )

3

10 [(0.02127 0.02631) 2(0.01562 0.01639) 4(0.01724 0.01538 0.01923)]3

1.0633

hds y y y y y y y

v


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SIMPSON'S RULE

Putting n = 3 in general formula


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Example:• Evaluate with n = 6 using Simpson’s 3/8 rule.Ans: Divide the interval (0,3) in to 6 equal part so,

By Simpson’s 3/8 rule,

3

01

dx

x

3 0 16 2

b ah

n

x 0 0.5 1 1.5 2 2.5 3

y 1 0.6667 0.5 0.4 0.3333 0.2857 0.25

3

0 6 1 2 3 4 5 30

3[( ) 3( ) 2 )

1 8dx h

y y y y y y y y x

13( )

2 [(1 0.25) 3(0.6667 0.5 0.3333 0.2857) 2(0.4)8

1.3888


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APPLICATIONS OF SIMPSON’S RULE

• If the various ordinates quadratures represent equispacedcross section areas, then Simpson’s rule give the volume of solid.

• As such, Simpson’s rule is very useful to Civil Engineers forCalculating the amount of earth that must be move to fill adepression or make a dam.

• Similarly if the ordinates denote at equal intervals of time,the Simpson’s rule gives the distance travelled.


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THANK YOU

Numerical Integration_Misal Gandhi

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