ࡱ> [!bjbj 4ΐΐNF8;W|̋6h ;(;(;(;(IcIcIcVIcobEcIcIc;(;(4:#x#x#xIc t;({R;(#xIc#x#x|";(p'6nR׊0̋ Ku$(IcIc#xIcIcIcIcIcYwIcIcIc̋IcIcIcIcIcIcIcIcIcIcIcIcIc :!INTEGRAL LIPAT TIGA  Bentuknya :  EMBED Equation.3  f (x,y,z) didefiniskan pada ruang tertutup V, V dibagi atas paralelepipedum tegak lurus oleh bidang-bidang sejajar bidang koordinat. Paralelepipedum dalam V kita beri nomor 1 sampai n, Paralelepipedum ke-i mempunyai volume iV. Integral tripel (integral lipat tiga) diperoleh dari limit dari jumlah :  EMBED Equation.3 = lim  EMBED Equation.3 (xi*, yi*, zi*) iV Jika n ( ", sedang diagonal maksimum dari iV ( 0, titik (xi*, yi*, zi*) dipilih sembarang dalam paralelepipedum ke-i Adanya suatu limit yang unik dapat ditunjukkan, jika f (x, y, z) kontinu di V. Teori sederhana berlaku untuk ruang tertutup V yang dilukiskan sebagai berikut : x1 < x < x2 , y1 (x) < y < y2 (x) dan z1 (x,y) < z < z2 (x,y), sehingga :  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  f (x,y,z) dz dy dx Integral ini dapat dianggap = volume dalam R4 = hypervolume Jika f (x,y,z) = 1 maka  EMBED Equation.3  adalah merupakan volume dari V Jika f = massa jenis benda, maka M = massa benda =  EMBED Equation.3  CONTOH SOAL Hitunglah  EMBED Equation.3 dengan f (x,y,z) = x + y + z dan R daerah dibatasi oleh x+y+z = a, (a>0), x=0, y=0, z=0. Jawab :  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  x + y + z dz dy dx = =  EMBED Equation.3  EMBED Equation.3  2. Hitunglah  EMBED Equation.3  dimana f (x, y, z) = x + y2 + z dan V dibatasi oleh x + y + z = 5, x = 0, y = 0 dan z = 0 Jawab :  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  x + y + z dz dy dx =  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  dy dx =  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  dy dx =  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  dy dx =  EMBED Equation.3   EMBED Equation.3  dx =  EMBED Equation.3   EMBED Equation.3  dx =  EMBED Equation.3   EMBED Equation.3  dx =  EMBED Equation.3  EMBED Equation.3  = =  EMBED Equation.3  Hitunglah volume dari R yang dibatasi oleh silinder z = 4-x dengan bidang-bidang x = 0, y = 0, y = 6 dan z = 0 Jawab :  V =  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  dz dy dx =  EMBED Equation.3   EMBED Equation.3  (4-x) dy dx =  EMBED Equation.3   EMBED Equation.3  dx = 6  EMBED Equation.3   EMBED Equation.3  dx = 6 .  EMBED Equation.3  = 6. ( 8 -  EMBED Equation.3 ) = 6.  EMBED Equation.3  =  EMBED Equation.3  = 32 4. Tentukan volume benda yang dibatasi oleh permukaan z = 9 - x - y dan bidang z =0 Jawab : V = 4  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  dz dx dy = 4  EMBED Equation.3   EMBED Equation.3  9 - x - y 2 dx dy = 4  EMBED Equation.3  9x -  EMBED Equation.3 x3 - yx  EMBED Equation.3  dy = 4  EMBED Equation.3  9  EMBED Equation.3  -  EMBED Equation.3 (9 - y)3/2 - y (9 - y)1/2 dy Misal y = 3 sin  dy = 3 cos  d  = 4  EMBED Equation.3  = 4  EMBED Equation.3  = 4  EMBED Equation.3  = 4 .  EMBED Equation.3  =  EMBED Equation.3  Tentukan volume benda di oktan pertama yang dibatasi oleh paraboloida z = x + y, tabung x + y = 4 dan bidang-bidang koordinat  Jawab : V =  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  dz dx dy =  EMBED Equation.3   EMBED Equation.3  x + y dx dy =  EMBED Equation.3   EMBED Equation.3 x3 - yx  EMBED Equation.3  dy =  EMBED Equation.3   EMBED Equation.3 (4-y) EMBED Equation.3 + y (4-y)1/2 dy Misal y = 2 sin  =  EMBED Equation.3  EMBED Equation.3  2 cos  d  =  EMBED Equation.3  EMBED Equation.3  cos 4 +  EMBED Equation.3 16 sin  cos  d  =  EMBED Equation.3  .  EMBED Equation.3   EMBED Equation.3  + 16 .  EMBED Equation.3  .  EMBED Equation.3  - 16  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3 + 4 EMBED Equation.3  - 3 EMBED Equation.3  = 2 EMBED Equation.3  Tentukan pusat massa daerah yang dibatasi oleh silinder parabolik z = 4 - x dan bidang-bidang x = 0, y = 0, y = 6 dan z = 0 dengan mengandaikan bahwa rapat massanya tetap sebesar  EMBED Equation.3  Jawab : Daerahnya lihat gambar dari soal 3 Volume =  EMBED Equation.3  = = 32 M = Massa total = jumlah massa =  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  dz dy dx = 32  EMBED Equation.3  Myz = Jumlah momen terhadap bidang yz =  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  x dz dy dx =  EMBED Equation.3   EMBED Equation.3 x (4 - x) dy dx = = 24  EMBED Equation.3  Mxz = Jumlah momen terhadap bidang xz =  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  y dz dy dx =  EMBED Equation.3   EMBED Equation.3 y (4 - x) dy dx = = 96  EMBED Equation.3  Mxy = Jumlah momen terhadap bidang xy =  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  z dz dy dx =  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  (4 - x) dy dx = =  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  = 3  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Jadi pusat massa adalah ( EMBED Equation.3 , 3,  EMBED Equation.3 ) SOAL-SOAL LATIHAN 1.  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EMBED Equation.3  2x  y  z dz dy dx = & (  EMBED Equation.3  2.  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  z  EMBED Equation.3  sin  dz dp d = & (  EMBED Equation.3  3.  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  x y z dz dy dx = (  EMBED Equation.3  4.  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3 4 sin  dp d d = & ( 2500  EMBED Equation.3  5. Hitung volume benda yang dibatasi oleh x + y = a, paraboloida z = x + y dan bidang z = 0 (  EMBED Equation.3  a4 6. 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