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8/14/2019 7. Integrasi
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8/14/2019 7. Integrasi
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8/14/2019 7. Integrasi
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Bina Nusantara
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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