INTEGRAL
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Transcript of INTEGRAL
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INTEGRAL
Kalkulus Teknik Informatika
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PENDAHULUAN
INTEGRAL DIFERENSIAL
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Contoh Integral
Temukan anti turunan dari Dari teori derivarif kita tahu
34)( xxf 4)( xxF
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Teorema A : Aturan Pangkat
Jika r adalah sembarang bilangan rasional kecuali
(-1), maka :
Jika r = 0 ? Perhatikan bahwa untuk anti derivatif suatu pangkat
dari x kita tambah pangkatnya dengan 1 dan membaginya dengan pangkat yg baru.
Anti turunan sering disebut dengan Integral Tak Tentu Dalam notasi disebut tanda integral,
sedangkan f(x) disebut integran
Cxdxx rr
r
11
1
,)( dxxf
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Teorema B : Kelinearan integral tak tentu
Andaikan f dan g mempunyai anti turunan (integral tak tentu) dan k adalah konstanta, maka
1. k f(x) dx = k f(x) dx
2. [ f(x) + g(x) ] dx = f(x) dx + g(x) dx
3. [ f(x) - g(x) ] dx = f(x) dx - g(x) dx
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Teorema C Aturan pangkat yang diperumum
Cxgdxxgxg rr
r
11
1 )]([)(')]([
Andaikan g suatu fungsi yang dapat didiferensialkan dan r suatu bil rasional bukan (-1), maka :
Contoh : Carilah integral dari f(x) sbb.
dxxxx )34()3( 3304
dxxx cossin30
1,11
1 rCuduu r
rr
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Integral Tentu
Teorema Kalkulus yg penting
Jika fungsi f(x) kontinu pada interval
a ≤ x ≤ b, maka
dimana F(x) adalah integral dari fungsi f(x)
pada a ≤ x ≤ b.
b
a
aFbFdxxf )()()(
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Contoh
Solusi
=
=
=
dxxx 1
2
3 3
1
2
24
2
3
4
xx
642
3
4
1
4
18
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Contoh
Solusi
=
= 14-13 = 11
dxxx 2
1
2 123
2123 xxx
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Contoh:Carilah area dibawah kurva dari fungsi berikut ini
Solusi
3
1
2 1 dxxA
dxx 3
1
2 1
3
1
3
3
x
x
1
3
139
3
412 67.10
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Grafik
1)( 2 xxf
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Area diantara dua kurva
Area diantara 2 kurva f(x) dan g(x)
b
a
dxxgxfA )()(
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Contoh Carilah area R yang berada diantara kurva dan
kurva
Solusi Carilah titik pertemuan antara 2 kurva
=> => x=1 or x=0
=> = = =
3xy 2xy
23 xx 012 xx
1
0
23 dxxxA
1
0
34
34
xxA
3
1
4
1
12
1
12
1
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Contoh Carilah area yang dibatasi oleh garis dan kurva
SolusiCarilah titik pertemuan:
23 3xxy
xy 4
xxx 43 23
0432 xxx
014 xxx
1,4,0 xxx
1
0
230
4
23 4343 dxxxxdxxxxA
1
0
2340
4
234
24
24
xxx
xxx
A
21
4
1)4(2)4(4
4
10 234A
4
332 A
4
332A
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Sifat-sifat Integral Tentu
INTEGRAL
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Sifat-sifat Integral Tentu
INTEGRAL
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Volume Benda Putar
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Metode Cakram
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Metode Cakram
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Metode Cakram
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Metode Cakram
TURUNAN DAN DIFERENSIAL
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Contoh 1
TURUNAN DAN DIFERENSIAL
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Contoh 2
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Metode Kulit Tabung
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Metode Kulit Tabung
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Metode Kulit Tabung
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Metode Kulit Tabung
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Contoh
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Latihan
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Integral Parsial 30
Integral Partial
Berdasarkan pada pengitegralan rumus turunan hasil dua kali fungsi :
Jika u dan v adalah fungsi x yang dapat dideferensiasi :
d(uv) = udv + vdu
udv = d(uv) – vdu
vduuvudv
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Integral Parsial 31
Aturan yg hrs diperhatikan
1. Bagian fungsi yang dipilih sebagai dv harus dapat segera diintegrasikan
2. tidak boleh lebih sulit daripada vdu udv
Contoh 1 :
xdxx cos
a. Misal : u = x dv = cos x dx
du = dx v = sin x
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Integral Parsial 32
Rumus integralnya :
xdxxxdxxx sinsincos
= x sin x + cos x + cb. Misal diambil :
u = cos x dv = x dx
du = -sin x dx v = x2/2Rumus Integral Parsialnya :
)sin(22
)(coscos22
dxxxx
xdxxx
Integralnya lebih susah
u dv u v - v du
Penting Sekali pemilihan u dan v
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Integral Parsial 33
Pengintegralan Parsial Berulang
Seringkali ditemui pengintegralan parsial berulang beberapa kali
xdxx sin2
Misal : u = x2 dv = sin x dx
du = 2x dx v = -cos x
Maka :
xdxxxxxdxx cos2cossin 22
- Tampak bahwa pangkat pada x berkurang
- Perlu pengintegralan parsial lagi
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Integral Parsial 34
Dari contoh 1 :
)cossin(2cossin 22 cxxxxxxdxx
= -x2cos x + 2x sinx + 2 cos x + K
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Integral Parsial 35
Contoh 3 :
xdxex sin
xdxexexdxe xxx coscoscos
Misal : u = ex dan dv = sinx dx
du = exdx dan v = - cosx
Maka :
Perlu penerapan integral parsial dalam integral kedua
xdxex cos u = ex dv = cos x dx
du = exdx v = sin x
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Integral Parsial 36
Sehingga :
xdxexexdxe xxx sinsincosBila hasil ini disubstitusikan pada hasil pertama
xdxexexexdxe xxxx sinsincossin
Cxexexdxe xxx sincossin2
Kxxexdxe xx )sin(cos2
1sin