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NumericalLecture 33Lecture 33

Analysis

Numerical

Analysis

Chapter 7

Numerical

Differentiation

and

Integration

Chapter 7

Numerical

Differentiation

and

Integration

NEWTON-COTES

INTEGRATION FORMULAE THE TRAPEZOIDAL RULE

( COMPOSITE FORM )

SIMPSON’S RULES

( COMPOSITE FORM )

ROMBERG’S INTEGRATIONDOUBLE INTEGRATION

Basic Issues in IntegrationWhat does an integralrepresent?

=

AREA^ =

VOLUME

b a

f (x) dx 

d b c a

g(x, y) dx dy

NUMERICALINTEGRATIONConsider the definite integral

b x^

a

I

f

x

d x





where

f

(x)

is

known

either

explicitly or is given as a tableof

values

corresponding

to

some

values

of

x,

whether

equispaced or not. Integrationof

such

functions

can

be

carried

out

using

numerical

techniques.

This area is approximated bythe

trapezium

formed

by

replacing the curve with itssecant

line

drawn

between

the

end

points

(x

0

,^

y

0

)

and

(x

1

, y

1

).

(^10)

0

0

1

1

3

0

1

( )

Error

(

)

(

)

2

12

x x^

f^

x dx

c y

c y

h

h

y

y

y

















xn^

= b

xn-

x^3

x^2

x^1

x= a^0

X

Y O

(x^2

, y 2

)

(x^0

, y 0

)

y

y

y

y = f(x)

Thus

Simpson’s

1/

rule

is

based

on

fitting

three

points

with a quadratic.Similarly,

for

n

=

3,

the

integration is found to be

This

is

known

as

Simpson’s

3/

rule,

which

is

based

on

fitting four points by a cubic.Still

higher

order

Newton-

Cotes integration formulae canbe derived for large values ofn.

The Trapezoidal Rule(Composite Form)The Newton-Cotes formula isbased on approximatingy = f (x) between (x

0

, y

0

) and

(x

1

, y

1

)

0

0

1

2

1

(

) (

2

2

2

2

)

n x x

n

n

n

f

x d x

h

y

y

y

y

y

E





















docsity.com

SIMPSON’S 1/3 RULE

(^20)

5

(^

)

0

1

2

x x

iv

I

f^

x dx

h

h

y

y

y

y



