Numerical integration

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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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
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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?
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Integration In Graphical Representation
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Mathematical Background of Integration
Table.1 Some Simple
Integral forms
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Numerical Integration
Newton-Cotes Integration Formulas
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FIGURE 2 The Integration by the area under (a) a single straight
line (b) a single parabola
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Newton-Cotes Integration available in Closed and Open Form
FIGURE 4 The difference between (a) closed
(b) open integration formula.
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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)
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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.
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Error of the Trapezoidal Rule
Then, the result can be written as
and
(3)
(4)
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The Multiple Application Trapezoidal Rule
(5)
It’s error is calculated by
(6)
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The Multiple Application Trapezoidal Rule
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Simpson’s Rules
There are two types of Simpson’s Rules:
 Simpson’s 1/3 Rule
 Simpson’s 3/8 Rule
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Simpson’s Rules
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Simpson’s 1/3 Rule
Then,
It’s error is estimated by
(7)
(8)
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The Multiple Application Simpson’s 1/3 Rules
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(9)
(10)
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and
Then (11)
(12)
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THANK YOU
BY: MOHAMMED ABDULQADER 28

Numerical integration

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