numerical integration

By: Mohammed Abdulqader 1

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 signx is called the variable of

integrationIntegrand

(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 integrationformula. It corresponds of the case where the polynomial in Eq.(1) isfirst-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



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Simpson’s Rules


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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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