This seminar is a bout Numerical Integration. In which most of methods are presented with examples.

1. By: Mohammed Abdulqader 1

2. History of integration
 Archimedes is the founder of surface areas and
volumes of solids such as the sphere and the cone. His
integration method was very modern.
 Gauss was the first to make graphs of integrals.
 Leibniz and Newton discovered calculus and found that
differentiation and integration undo each other
By: Mohammed Abdulqader
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3. How Integration Applies To The Real World
 Integration was used to design the
PETRONAS Towers making it stronger
 Many differential equations were used in the
designing of the Sydney Opera House
 Finding areas under curved surfaces,
Centers of mass, displacement and Velocity,
and fluid flow are other uses of integration
By: Mohammed Abdulqader
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4. Integration is the process of evaluating an indefinite integral
or a definite integral
Integral sign
x is called the variable of
integration
Integrand
(1)
What is the Integration?
By: Mohammed Abdulqader
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5. Integration In Graphical Representation
By: Mohammed Abdulqader
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6. Mathematical Background of Integration
Table.1 Some Simple
Integral forms
By: Mohammed Abdulqader
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7. Numerical Integration
Newton-Cotes Integration Formulas
By: Mohammed Abdulqader
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8. FIGURE 2 The Integration by the area under (a) a single straight
line (b) a single parabola
Numerical Integration
Newton-Cotes Integration Formulas
By: Mohammed Abdulqader
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9. Numerical Integration
Newton-Cotes Integration Formulas
By: Mohammed Abdulqader
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10. Newton-Cotes Integration available in Closed and Open Form
FIGURE 4 The difference between (a) closed
(b) open integration formula.
By: Mohammed Abdulqader
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11. The Trapezoidal Rule
The trapezoidal rule is the first of the Newton-Cotes integration
formula. It corresponds of the case where the polynomial in Eq.(1) is
first-order
This is called the Trapezoidal Rule (2)
By: Mohammed Abdulqader
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12. The Trapezoidal Rule is equivalent to approximating the area of the
trapezoid under the straight line connecting f(a) and f(b) in Fig. 5.
The Trapezoidal Rule
By: Mohammed Abdulqader
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13. Error of the Trapezoidal Rule
Then, the result can be written as
and
(3)
(4)
By: Mohammed Abdulqader
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14. The Trapezoidal Rule
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15. The Multiple Application Trapezoidal Rule
(5)
It’s error is calculated by
(6)
By: Mohammed Abdulqader
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16. The Multiple Application Trapezoidal Rule
By: Mohammed Abdulqader
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17. Simpson’s Rules
There are two types of Simpson’s Rules:
 Simpson’s 1/3 Rule
 Simpson’s 3/8 Rule
By: Mohammed Abdulqader
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18. Simpson’s Rules
By: Mohammed Abdulqader
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19. Simpson’s 1/3 Rule
Then,
It’s error is estimated by
(7)
(8)
By: Mohammed Abdulqader
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20. Simpson’s 1/3 Rule
By: Mohammed Abdulqader
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21. The Multiple Application Simpson’s 1/3 Rules
By: Mohammed Abdulqader
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22. The Multiple Application Simpson’s 1/3 Rules
It’s error is estimated by
(9)
(10)
By: Mohammed Abdulqader
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23. The Multiple Application Simpson’s 1/3 Rules
By: Mohammed Abdulqader
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24. Simpson’s 3/8 Rule
and
Then (11)
(12)
It’s error is estimated by
By: Mohammed Abdulqader
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25. Simpson’s 3/8 Rule
By: Mohammed Abdulqader
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26. Simpson’s 3/8 Rule
By: Mohammed Abdulqader
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27. Simpson’s 3/8 Rule
By: Mohammed Abdulqader
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28. THANK YOU
BY: MOHAMMED ABDULQADER 28