Unit5 Numerical Integration 2140706 Numerical Statistical Methods Humanities

Unit-5 Numerical Integration 2140706 – Numerical & Statistical Methods Humanities & Science Department Numerical & Statistical methods (2140706) Darshan Institute Of Engineering & Technology

What is Integration ? Ø Integral Sign Integrand Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 2

Graphical representation of Integral Numerical & Statistical methods (2140706) Darshan Institute Of Engineering & Technology 3

Integration Ø Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 4

Why Numerical Integration ? Ø Solve it, if you can Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 5

Numerical Integration Consider the observation table Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 6

Methods of Numerical integration 1. Trapezoidal Rule 2. Simpson’s 1/3 Rule 3. Simpson’s 3/8 Rule 4. Weddle’s Rule Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 7

Trapezoidal Rule “Trapezoid” approximation with one subinterval One interval trapezoidal rule Numerical & Statistical methods (2140706) Darshan Institute Of Engineering & Technology 8

Trapezoidal Rule “Trapezoid” approximation with two subinterval Two interval trapezoidal rule Numerical & Statistical methods (2140706) Darshan Institute Of Engineering & Technology 9

Trapezoidal Rule “Trapezoid” approximation with three subinterval Three interval trapezoidal rule Numerical & Statistical methods (2140706) Darshan Institute Of Engineering & Technology 10

Derive Trapezoidal rule.

Trapezoidal Rule Ø Consider the observation table Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 13

Example Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 14

1. 3499 Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 15

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Example Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 17

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Example Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 19

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Simpson’s 1/3 Rule Trapezoidal rule fit two points , Here Parabola is fitted through three points Numerical & Statistical methods (2140706) Darshan Institute Of Engineering & Technology 22

Simpson’s 1/3 Rule Parabola 0 Numerical & Statistical methods (2140706) Darshan Institute Of Engineering & Technology 23

Simpson’s 1/3 Rule Numerical & Statistical methods (2140706) Darshan Institute Of Engineering & Technology 24

Simpson’s 1/3 Rule ü The no. of ordinates ( y values ) is odd. ü The accuracy can be improved by increasing n(subinterval) ü Integration is approximated by second order polynomial (quadratic). Numerical & Statistical methods (2140706) Darshan Institute Of Engineering & Technology 25

Simpson’s 1/3 Rule Ø Consider the observation table Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 26

Example 1 Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 27

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Example Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 30

Example 5 T (time) 0 12 V (Speed) 0 3. 6 24 36 10. 08 18. 90 48 21. 6 60 72 18. 54 10. 26 84 96 108 120 4. 5 5. 4 9 Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 31

Example 5 Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 32

t 0 12 v 0 3. 6 Ø 24 36 48 10. 08 18. 90 60 21. 6 72 18. 54 10. 26 84 96 108 120 4. 5 5. 4 9 Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 33

Example 8 Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 34

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Simpson’s 3/8 Rule Ø Consider the observation table Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 37

Simpson’s 3/8 Rule a b c Simpson’s 3/8 Rule d a b c Simpson’s 1/3 Rule Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 38

Error bounds in Simpson's 3/8 rule Ø Constant Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 39

Example 2 Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 40

1 Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 41

1 Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 42

Example Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 43

Ø Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 44

Ø Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 45

Example Ø x y 4 4. 2 4. 4 4. 6 4. 8 5. 0 5. 2 1. 3863 1. 4351 1. 4816 1. 5261 1. 5686 1. 6094 1. 6487 Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 46

Ø Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 47

Example Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 48

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Weddle’s rule How to remember it easily ? We remember it to make formula. Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 51

Weddle’s rule Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 52

Example -1 Ø X 25 25. 1 25. 2 25. 3 25. 4 25. 5 25. 6 F(x) 3. 205 3. 217 3. 232 3. 245 3. 256 3. 268 3. 28 Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 53

X 25 25. 1 25. 2 25. 3 25. 4 25. 5 25. 6 F(x) 3. 205 3. 217 3. 232 3. 245 3. 256 3. 268 3. 28 Ø Numerical and statistical method (2140706) Darshan Institute of engineering & Technology 54