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

By: Mohammed Abdulqader

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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?

By: Mohammed Abdulqader

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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

Numerical Integration

Newton-Cotes Integration Formulas

By: Mohammed Abdulqader

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Numerical Integration

Newton-Cotes Integration Formulas

By: Mohammed Abdulqader

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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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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

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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.

The Trapezoidal Rule

By: Mohammed Abdulqader

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Error of the Trapezoidal Rule

Then, the result can be written as

and

(3)

(4)

By: Mohammed Abdulqader

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The Trapezoidal Rule

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The Multiple Application Trapezoidal Rule

(5)

It’s error is calculated by

(6)

By: Mohammed Abdulqader

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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

By: Mohammed Abdulqader

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Simpson’s Rules

By: Mohammed Abdulqader

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Simpson’s 1/3 Rule

Then,

It’s error is estimated by

(7)

(8)

By: Mohammed Abdulqader

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Simpson’s 1/3 Rule

By: Mohammed Abdulqader

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The Multiple Application Simpson’s 1/3 Rules

By: Mohammed Abdulqader

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The Multiple Application Simpson’s 1/3 Rules

It’s error is estimated by

(9)

(10)

By: Mohammed Abdulqader

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The Multiple Application Simpson’s 1/3 Rules

By: Mohammed Abdulqader

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Simpson’s 3/8 Rule

and

Then (11)

(12)

It’s error is estimated by

By: Mohammed Abdulqader

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Simpson’s 3/8 Rule

By: Mohammed Abdulqader

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Simpson’s 3/8 Rule

By: Mohammed Abdulqader

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Simpson’s 3/8 Rule

By: Mohammed Abdulqader

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THANK YOU

BY: MOHAMMED ABDULQADER 28