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Numerical Integration - Tools in Mechanical Engineering - Lecture Slides, Slides of Mechanical Engineering

AMET UniversityMechanical Engineering

These are the Lecture Slides of Tools in Mechanical Engineering which includes Steady Heat Equation, Bounday Conditions, Fourier Sine Coefficients, Original Variable, Temperature Distribution, Constant Temperature, Contour Plot, New Variable etc.Key important points are:Numerical Integration, Definite Integration, Discretized Points, Weighted Factor, Integration Schemes, Rectangular Rule, Trapezoidal Rule, Simpson’s Rule, Perfect Estimation, Subdivisions

Typology: Slides

2012/2013

Uploaded on 03/27/2013

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Numerical Integration
•In general, a numerical integration is the approximation
of a definite integration by a “weighted” sum of function
values at discretized points within the interval of
integration.
0
( ) ( )
where is the weighted factor depending on the integration
schemes used, and ( ) is the function value evaluated at the
given point
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b
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f x dx w f x
w
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x
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Numerical Integration

  • In general, a numerical integration is the approximation

of a definite integration by a “weighted” sum of function

values at discretized points within the interval of

integration.

0

where is the weighted factor depending on the integration

schemes used, and ( ) is the function value evaluated at the

given point

b^ N a i^ i i i i i

f x dx w f x

w

f x

x

^ ^ 

Rectangular Rule

x=a (^) x=b

Approximate the integration, , that is the area under the curve by a series of rectangles as shown. The base of each of these rectangles is Dx =(b-a)/n and its height can be expressed as f(xi*) where xi* is the midpoint of each rectangle

 a^ b f^ ( ) x dx

x=x 1 * (^) x=xn*

height=f(x 1 ) height=f(xn)

1 2

1 2

( ) ( *) ( *) .. ( *)

[ ( *) ( *) .. ( *)]

b a n

n

f x dx f x x f x x f x x

x f x f x f x

 D  D  D

 D  

f(x)

x

Simpson’s Rule

Still, the more accurate integration formula can be achieved by approximating the local curve by a higher order function, such as a quadratic polynomial. This leads to the Simpson’s rule and the formula is given as:

1 2 3

2 2 2 1

( ) [ ( ) 4 ( ) 2 ( ) 4 ( ) ..

..2 ( ) 4 ( ) ( )]

b a m m

x f x dx f a f x f x f x

f x (^)  f x (^)  f b

D

It is to be noted that the total number of subdivisions has to be an even number in order for the Simpson’s formula to work properly.

Examples

3

(^2 3 4 2 4 ) 1 1

Integrate ( ) between 1 and 2.

1 1 x dx= | (2 1 ) 3. 4 4 2- Using 4 subdivisions for the numerical integration: x= 0. 4 Rectangular rule:

f x x x x

x

D 

i xi* f(xi*) 1 1.125 1. 2 1.375 2. 3 1.625 4. 4 1.875 6.

(^2 ) 1 [ (1.125) (1.375) (1.625) (1.875)] 0.25(14.9) 3.

x dx  D x ffff  