Numerical integration
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This document discusses the history and applications of integration. It provides an overview of how integration was developed over time by mathematicians like Archimedes, Gauss, Leibniz, and Newton. It also outlines real-world uses of integration in engineering projects like designing the PETRONAS Towers and Sydney Opera House. The document then explains numerical integration methods like the Trapezoidal Rule, Simpson's Rule, and their variations. It provides formulas and examples of how to apply these rules to approximate definite integrals.
Numerical integration
- 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 2
- 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 3
- 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 4
- 5. Integration In Graphical Representation By: Mohammed Abdulqader 5
- 6. Mathematical Background of Integration Table.1 Some Simple Integral forms By: Mohammed Abdulqader 6
- 7. Numerical Integration Newton-Cotes Integration Formulas By: Mohammed Abdulqader 7
- 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 8
- 9. Numerical Integration Newton-Cotes Integration Formulas By: Mohammed Abdulqader 9
- 10. Newton-Cotes Integration available in Closed and Open Form FIGURE 4 The difference between (a) closed (b) open integration formula. By: Mohammed Abdulqader 10
- 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 11
- 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 12
- 13. Error of the Trapezoidal Rule Then, the result can be written as and (3) (4) By: Mohammed Abdulqader 13
- 14. The Trapezoidal Rule By: Mohammed Abdulqader 14
- 15. The Multiple Application Trapezoidal Rule (5) It’s error is calculated by (6) By: Mohammed Abdulqader 15
- 16. The Multiple Application Trapezoidal Rule By: Mohammed Abdulqader 16
- 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 17
- 18. Simpson’s Rules By: Mohammed Abdulqader 18
- 19. Simpson’s 1/3 Rule Then, It’s error is estimated by (7) (8) By: Mohammed Abdulqader 19
- 20. Simpson’s 1/3 Rule By: Mohammed Abdulqader 20
- 21. The Multiple Application Simpson’s 1/3 Rules By: Mohammed Abdulqader 21
- 22. The Multiple Application Simpson’s 1/3 Rules It’s error is estimated by (9) (10) By: Mohammed Abdulqader 22
- 23. The Multiple Application Simpson’s 1/3 Rules By: Mohammed Abdulqader 23
- 24. Simpson’s 3/8 Rule and Then (11) (12) It’s error is estimated by By: Mohammed Abdulqader 24
- 25. Simpson’s 3/8 Rule By: Mohammed Abdulqader 25
- 26. Simpson’s 3/8 Rule By: Mohammed Abdulqader 26
- 27. Simpson’s 3/8 Rule By: Mohammed Abdulqader 27
- 28. THANK YOU BY: MOHAMMED ABDULQADER 28
Editor's Notes
- #16: Ed by