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INTEGRASI NUMERIKPengertian:

Secara matematika merupakan anti derivative

Secara praktis integral suatu fungsi pada batas tertentu dapat

diartikan sebagai luas daerah dibawah kurva dengan batas

yang ditentukan dan sumbu x

!!

dxxgxIxgdx

xdI

)()()(

)(

!2

1

)()(

x

x

dxxgxIx

g(x)

x1 x2

y

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Numerical Integration

Often g(x) cannot be integrated analytically to give aclosed form

expression forI(x), e.g. solution to y =(1+xy).

Motivates need to find approximate solutions forI(x).

In the differentiallimit, an integral is equivalent to a summation

operation:

Approximate methods for determining integrals are mostly basedon idea ofarea between integrand and axis.

Area determined by the summation of the areas of strips of

finite width (x.

Particular method depends on how strips are determined.

(! p(2

1

2

1

)(lim)(0

x

xx

x

x

xxgdxxg

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Numerical Integration by Rectangular Method

g(x) is approximated byaconstant value over the finite distance (x.

Areas of rectangles summed to give approximation of integral.

x

g(x)

x1 x2

))()()(()( 211

2

1

xxgxxgxgxdxxg

x

x

(((}

-

n

xxx 12

!(

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Integration as a Weighted Sum of Sample Function Evaluations

Think of integration as the weighted sum of function evaluations at

a number of discrete sample points, c.f. interpolation.

e.g. rectangular method:

Weights for rectangular method sum to give integration interval:

)()()()( 21211

2

1

xxgwxxgwxgwdxxg n

x

x

((}

-

xwwwn

(!!! -21

12xxxn !(

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Contoh

Hitung untuk n=8, maka1

0

2

dxex 125,08

)10(!

!(x

36232,1)...(125,0)()875,0()25,0()125,0(0 22

!$ eeeefI

Latihan:

Menggunakan Metode Rectangular hitung hasil integral di atas

untuk n=10

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Formula Trapezoida Komposit dan Error

Rumus Komposit

Error

!

!1

1

21)](2)()()[

2()(

n

j

jxfxfxfh

fT

)('')()12

1()(

12

2 FfxxhfRT !

Dimana: x1 F x2

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Contoh:

1

0

2

dxex

!

!1

1

21)](2)()()[

2()(

n

j

jxfxfxfhfT

125,08

)10(!

!h

)]...(2[)(2222)875,0()375,0()25,0()125,0(10

2

125,0eeeeeefI !

Hitung Untuk n=8

I(f)= 1,469712

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Metode Simpson

Simpsons method top of each pair of strips varies quadratically :

Weighted sum of function evaluations at sample points.

(

(

((((

})()

2(4)(2

)2

3(4)(2)2

(4)(

6)(

222

11112

1 xgx

xgxxg

xxgxxgxxgxgx

dxxg

x

x -

Exact for

polynomials up to

order3

x

g(x)

Weights sum to give x2-x1.

(xEvaluate in centre

also 3 points to get

polynomial

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Rumus Simpsons KompositIntegral dari

)()()( fsdxxffIb

a}!

!

!

!

m

j

j

m

j

xj xfxfbfafhfS1

12

1

1

)](4)(2)()()[3

()(

Error: Rs(f)=-(h4/180)(b-a)f(4)(F)

Dimana n = 2m, h=(b-a)/n, aF b

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