06 intergral reimann
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Transcript of 06 intergral reimann
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INTEGRAL TERTENTU(RIEMANN)
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• Integral tertentu (Riemann)
Integral tertentu digunakan untuk menghitung luas area yang terletak diantara y=f(x) dan sumbu horisontal x, dalam suatu rentangan wilayah yang dibatasi oleh x=a dan x=b.
• Integral tidak tertentu
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• If there are n slices, each slice will have width
• The interval [a,b] will be partitioned into n subintervals
[xo,x1], [x1,x2], …, [xn-1,xn]
where
xo=a, x1=a+Δx, x2=a+2Δx, …, xn=b
• The points xo, x1, …, xn are called partition points.
• On each subinterval [xk-1,xk], we form the rectangle of height f(xk-1).
• The kth rectangle will have area: f(xk-1)• Δx
nabx
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• The Riemann sum
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Contoh 1:
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Sifat Integral Riemann
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Kelinearan integral Riemann• Misalkan bahwa f dan g terintegralkan pada [a, b] dan k
konstanta, maka kf dan f = g terintegrasikan dan:
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Teori Dasar Kalkulus I
• Misal f(x) kontinu pada interval [a, b] dan F(x) disebut sebagai anti turunan dari f(x) sehingga:
maka:
dengan notasi lain:
)(' xfxF
x
a
dttfxF )()(
54
1
:
43 xxFmakaxxfjika
contoh
)()( xfdttfdx
d x
a
xfdttfdx
d
xfxFdx
d
xfxF
x
a
'
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Teori Dasar Kalkulus II
• Misal f(x) kontinu pada interval [a, b] dan F(x) suatu anti turunan dari f(x), maka:
f x dx F b F aa
b( ) ( ) ( )
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Contoh:
Perlihatkan bahwa: 22
22 abdxx
b
a
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Teorema Nilai Rata-rata untuk Integral
• Jika f fungsi kontinu pada [a, b] maka terdapat sebuah bilangan c pada [a, b] sedemikian sehingga:
atau
f (c) merupakan nilai rata-rata integral dari f(x)
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Contoh:
Tentukan nilai rata-rata fungsi f(x) = x2 + 1 pada interval [1,3].