Numerical integration
-
Upload
mohammedaq -
Category
Presentations & Public Speaking
-
view
223 -
download
2
Transcript of Numerical integration
![Page 1: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/1.jpg)
By: Mohammed Abdulqader 1
![Page 2: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/2.jpg)
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
![Page 3: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/3.jpg)
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
![Page 4: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/4.jpg)
Integration is the process of evaluating an indefinite integral
or a definite integral
Integral signx is called the variable of
integrationIntegrand
(1)
What is the Integration?
By: Mohammed Abdulqader
4
![Page 5: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/5.jpg)
Integration In Graphical Representation
By: Mohammed Abdulqader
5
![Page 6: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/6.jpg)
Mathematical Background of Integration
Table.1 Some Simple
Integral forms
By: Mohammed Abdulqader
6
![Page 7: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/7.jpg)
Numerical Integration
Newton-Cotes Integration Formulas
By: Mohammed Abdulqader
7
![Page 8: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/8.jpg)
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
![Page 9: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/9.jpg)
Numerical Integration
Newton-Cotes Integration Formulas
By: Mohammed Abdulqader
9
![Page 10: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/10.jpg)
Newton-Cotes Integration available in Closed and Open Form
FIGURE 4 The difference between (a) closed
(b) open integration formula.
By: Mohammed Abdulqader
10
![Page 11: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/11.jpg)
The Trapezoidal Rule
The trapezoidal rule is the first of the Newton-Cotes integrationformula. It corresponds of the case where the polynomial in Eq.(1) isfirst-order
This is called the Trapezoidal Rule (2)
By: Mohammed Abdulqader
11
![Page 12: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/12.jpg)
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
![Page 13: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/13.jpg)
Error of the Trapezoidal Rule
Then, the result can be written as
and
(3)
(4)
By: Mohammed Abdulqader
13
![Page 14: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/14.jpg)
The Trapezoidal Rule
By: Mohammed Abdulqader
14
![Page 15: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/15.jpg)
The Multiple Application Trapezoidal Rule
(5)
It’s error is calculated by
(6)
By: Mohammed Abdulqader
15
![Page 16: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/16.jpg)
The Multiple Application Trapezoidal Rule
By: Mohammed Abdulqader
16
![Page 17: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/17.jpg)
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
![Page 18: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/18.jpg)
Simpson’s Rules
By: Mohammed Abdulqader
18
![Page 19: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/19.jpg)
Simpson’s 1/3 Rule
Then,
It’s error is estimated by
(7)
(8)
By: Mohammed Abdulqader
19
![Page 20: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/20.jpg)
Simpson’s 1/3 Rule
By: Mohammed Abdulqader
20
![Page 21: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/21.jpg)
The Multiple Application Simpson’s 1/3 Rules
By: Mohammed Abdulqader
21
![Page 22: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/22.jpg)
The Multiple Application Simpson’s 1/3 Rules
It’s error is estimated by
(9)
(10)
By: Mohammed Abdulqader
22
![Page 23: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/23.jpg)
The Multiple Application Simpson’s 1/3 Rules
By: Mohammed Abdulqader
23
![Page 24: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/24.jpg)
Simpson’s 3/8 Rule
and
Then (11)
(12)
It’s error is estimated by
By: Mohammed Abdulqader
24
![Page 25: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/25.jpg)
Simpson’s 3/8 Rule
By: Mohammed Abdulqader
25
![Page 26: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/26.jpg)
Simpson’s 3/8 Rule
By: Mohammed Abdulqader
26
![Page 27: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/27.jpg)
Simpson’s 3/8 Rule
By: Mohammed Abdulqader
27
![Page 28: Numerical integration](https://reader033.fdocuments.us/reader033/viewer/2022051016/55a6a1201a28ab27578b46d8/html5/thumbnails/28.jpg)
THANK YOU
BY: MOHAMMED ABDULQADER 28