132 Chu . o . ng 12. T´ıch phˆan h`am nhiˆe ` ubiˆe ´ n 51.  D ln(x 2 + y 2 ) x 2 + y 2 dxdy; D :1 x 2 + y 2  e.(DS. 2π) 52.  D (x 2 + y 2 )dxdy; D gi´o . iha . nbo . ’ ic´acd u . `o . ng tr`on x 2 + y 2 +2x − 1=0,x 2 + y 2 +2x =0. (DS. 5π 2 ) Chı ’ dˆa ˜ n. D ˘a . t x − 1=r cos ϕ, y = r sin ϕ. T´ınh thˆe ’ t´ıch cu ’ avˆa . tthˆe ’ gi´o . iha . nbo . ’ i c´ac m˘a . td ˜achı ’ ra. 53. x =0,y =0,z =0,x + y + z = 1. (D S. 1 6 ) 54. x =0,y =0,z =0,x + y =1,z = x 2 + y 2 .(DS. 1 6 ) 55. z = x 2 + y 2 , y = x 2 , y =1,z = 0. (DS. 88 105 ) 56. z =  x 2 + y 2 , x 2 + y 2 = a 2 , z = 0. (DS. 2 3 πa 3 ) 57. z = x 2 + y 2 , x 2 + y 2 = a 2 , z = 0. (DS. πa 4 2 ) 58. z = x, x 2 + y 2 = a 2 , z = 0. (DS. 4a 3 3 ) 59. z =4−x 2 − y 2 , x = ±1, y = ±1. (DS. 13 1 3 ) 60. 2 − x − y −2z =0,y = x 2 , y = x.(DS. 11 120 ) 61. x 2 + y 2 =4x, z = x, z =2x.(DS. 4π) T´ınh diˆe . n t´ıch c´ac phˆa ` nm˘a . td ˜achı ’ ra. 62. Phˆa ` nm˘a . t ph˘a ’ ng 6x +3y +2z = 12 n˘a ` m trong g´oc phˆa ` n t´am I. (D S. 14) 63. Phˆa ` nm˘a . t ph˘a ’ ng x + y + z =2a n˘a ` m trong m˘a . t tru . x 2 + y 2 = a 2 . (D S. 2a 2 √ 3) 12.2. T´ıch phˆan 3-l´o . p 133 64. Phˆa ` nm˘a . t paraboloid z = x 2 + y 2 n˘a ` m trong m˘a . t tru . x 2 + y 2 =4. (D S. π 6 (17 √ 17 − 1)) 65. Phˆa ` nm˘a . t2z = x 2 + y 2 n˘a ` m trong m˘a . t tru . x 2 + y 2 =1. (D S. 2 3 (2 √ 2 − 1)π) 66. Phˆa ` nm˘a . t n´on z =  x 2 + y 2 n˘a ` m trong m˘a . t tru . x 2 + y 2 = a 2 . (D S. πa 2 √ 2) 67. Phˆa ` nm˘a . tcˆa ` u x 2 +y 2 +z 2 = R 2 n˘a ` m trong m˘a . t tru . x 2 +y 2 = Rx. (D S. 2R 2 (π − 2)) 68. Phˆa ` nm˘a . t n´on z 2 = x 2 + y 2 n˘a ` m trong m˘a . t tru . x 2 + y 2 =2x. (D S. 2 √ 2π) 69. Phˆa ` nm˘a . t tru . z 2 =4x n˘a ` m trong g´oc phˆa ` n t´am th´u I v`a gi´o . iha . n bo . ’ im˘a . t tru . y 2 =4x v`a m˘a . t ph˘a ’ ng x = 1. (DS. 4 3 (2 √ 2 − 1)) 70. Phˆa ` nm˘a . tcˆa ` u x 2 + y 2 + z 2 = R 2 n˘a ` m trong m˘a . t tru . x 2 + y 2 = a 2 (a  R). (DS. 4πa(a − √ a 2 − R 2 )) 12.2 T´ıch phˆan 3-l´o . p 12.2.1 Tru . `o . ng ho . . pmiˆe ` n h`ınh hˆo . p Gia ’ su . ’ miˆe ` n D ⊂ R 3 : D =[a, b] × [c, d] ×[e, g]={(x, y, z):a  x  b, c  y  d, e  z  g} v`a h`am f(x,y, z)liˆen tu . c trong D. Khi d ´o t´ıch phˆan 3-l´o . pcu ’ a h`am f(x,y, z) theo miˆe ` n D d u . o . . c t´ınh theo cˆong th ´u . c  D f(x,y, z)dxdydz = b  a  d  c  g  e f(x,y, z)dz  dy  dx = b  a dx d  c dy g  e f(M)dx. (12.15) 134 Chu . o . ng 12. T´ıch phˆan h`am nhiˆe ` ubiˆe ´ n T`u . (12.15) suy ra c´ac giai d oa . n t´ınh t´ıch phˆan 3-l´o . p: (i) D ˆa ` u tiˆen t´ınh I(x, y)= g  e f(M)dz; (ii) Tiˆe ´ p theo t´ınh I(x)= d  c I(x, y)dy; (iii) Sau c`ung t´ınh t´ıch phˆan I = b  a I(x)dx. Nˆe ´ u t´ıch phˆan (12.15) d u . o . . c t´ınh theo th ´u . tu . . kh´ac th`ı c´ac giai d oa . n t´ınh vˆa ˜ ntu . o . ng tu . . :d ˆa ` u tiˆen t´ınh t´ıch phˆan trong, tiˆe ´ pdˆe ´ n t´ınh t´ıch phˆan gi˜u . a v`a sau c`ung l`a t´ınh t´ıch phˆan ngo`ai. 12.2.2 Tru . `o . ng ho . . pmiˆe ` n cong 1 + Gia ’ su . ’ h`am f(M)liˆen tu . c trong miˆe ` nbi . ch˘a . n D =  (x, y, z):a  x  b, ϕ 1 (x)  y  ϕ 2 (x),g 1 (x, y)  z  g 2 (x, y)  . Khi d ´o t´ıch phˆan 3-l´o . pcu ’ a h`am f(M) theo miˆe ` n D d u . o . . c t´ınh theo cˆong th´u . c  D f(M)dxdydz = b  a  ϕ 2 (x)  ϕ 1 (x)  g 2 (x,y)  g 1 (x,y) f(M)dx  dy  dx (12.16) ho˘a . c  D f(M)dxdydz =  D(x,y) dxdy g 2 (x,y)  g 1 (x,y) f(M)dz, (12.17) trong d ´o D(x, y)l`ah`ınh chiˆe ´ u vuˆong g´oc cu ’ a D lˆen m˘a . t ph˘a ’ ng Oxy. Viˆe . ct´ınh t´ıch phˆan 3-l´o . pd u . o . . c quy vˆe ` t´ınh liˆen tiˆe ´ p ba t´ıch phˆan thˆong 12.2. T´ıch phˆan 3-l´o . p 135 thu . `o . ng theo (12.16) t `u . t´ıch phˆan trong, tiˆe ´ pd ˆe ´ nt´ıch phˆan gi˜u . av`a sau c`ung l`a t´ınh t´ıch phˆan ngo`ai. Khi t´ınh t´ıch phˆan 3-l´o . p theo cˆong th ´u . c (12.17): d ˆa ` u tiˆen t´ınh t´ıch phˆan trong v`a sau d´o c ´o t h ˆe ’ t´ınh t´ıch phˆan 2-l´o . p theo miˆe ` n D(x, y) theo c´ac phu . o . ng ph´ap d ˜a c´o trong 12.1. 2 + Phu . o . ng ph´ap d ˆo ’ ibiˆe ´ n. Ph´ep dˆo ’ ibiˆe ´ n trong t´ıch phˆan 3-l´o . p d u . o . . ctiˆe ´ n h`anh theo cˆong th ´u . c  D f(M)dxdydz =  D ∗ f  ϕ(u, v, w),ψ(u, v, w),χ(u, v, w)  × ×    D(x, y, z) D(u, v, w)    dudvdw, (12.18) trong d ´o D ∗ l`a miˆe ` nbiˆe ´ n thiˆen cu ’ ato . adˆo . cong u, v, w tu . o . ng ´u . ng khi c´ac d iˆe ’ m(x, y, z)biˆe ´ n thiˆen trong D: x = ϕ(u, v, w), y = ψ(u, v, w), z = χ(u, v, w), D(x, y, z) D(u, v, w) l`a Jacobiˆen cu ’ a c´ac h`am ϕ, ψ, χ J = D(x, y, z) D(u, v, w) =            ∂ϕ ∂u ∂ϕ ∂v ∂ϕ ∂w ∂ψ ∂u ∂ψ ∂v ∂ψ ∂w ∂χ ∂u ∂χ ∂v ∂χ ∂w            =0. (12.19) Tru . `o . ng ho . . pd ˘a . cbiˆe . tcu ’ ato . adˆo . cong l`a to . adˆo . tru . v`a to . adˆo . cˆa ` u. (i) Bu . ´o . c chuyˆe ’ nt`u . to . ad ˆo . Dˆec´ac sang to . adˆo . tru . (r, ϕ, z)du . o . . c thu . . c hiˆe . n theo c´ac hˆe . th ´u . c x = r cos ϕ, y = r sin ϕ, z = z;0 r<+∞, 0  ϕ<2π, −∞ <z<+∞.T`u . (12.19) suy ra J = r v`a trong to . a d ˆo . tru . ta c´o  D f(M)dxdydz =  D ∗ f  r cos ϕ, r sin ϕ, z  rdrdϕdz, (12.20) trong d ´o D ∗ l`a miˆe ` nbiˆe ´ n thiˆen cu ’ ato . adˆo . tru . tu . o . ng ´u . ng khi d iˆe ’ m (x, y, z)biˆe ´ n thiˆen trong D. 136 Chu . o . ng 12. T´ıch phˆan h`am nhiˆe ` ubiˆe ´ n (ii) Bu . ´o . c chuyˆe ’ nt`u . to . ad ˆo . Dˆec´ac sang to . adˆo . cˆa ` u(r, ϕ, θ)du . o . . c thu . . chiˆe . n theo c´ac hˆe . th ´u . c x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ,0 r<+∞,0 ϕ<2π,0 θ  π.T`u . (12.19) ta c´o J = r 2 sin θ v`a trong to . adˆo . cˆa ` u ta c´o  D f(M)dxdydz = =  D ∗ f  r sin θ cos ϕ, r sin θ sin ϕ, r cos θ  r 2 sin θdrdϕdθ, (12.21) trong d ´o D ∗ l`a miˆe ` nbiˆe ´ n thiˆen cu ’ ato . adˆo . cˆa ` utu . o . ng ´u . ng khi d iˆe ’ m (x, y, z)biˆe ´ n thiˆen trong D. 12.2.3 Thˆe ’ t´ıch cu ’ avˆa . tthˆe ’ cho´an hˆe ´ tmiˆe ` n D ⊂ R 3 du . o . . c t´ınh theo cˆong th ´u . c V D =  D dxdydz. (12.22) 12.2.4 Nhˆa . n x´et chung B˘a ` ng c´ach thay dˆo ’ ith´u . tu . . t´ınh t´ıch phˆan trong t´ıch phˆan 3-l´o . ptas˜e thu d u . o . . c c´ac cˆong th´u . ctu . o . ng tu . . nhu . cˆong th´u . c (12.16) d ˆe ’ t´ınh t´ıch phˆan. Viˆe . c t`ım cˆa . n cho t´ıch phˆan d o . n thˆong thu . `o . ng khi chuyˆe ’ nt´ıch phˆan 3-l´o . pvˆe ` t´ıch phˆan l˘a . pd u . o . . c thu . . chiˆe . nnhu . d ˆo ´ iv´o . i tru . `o . ng ho . . p t´ıch phˆan 2-l´o . p. C ´ AC V ´ IDU . V´ı du . 1. T´ınh t´ıch phˆan l˘a . p I = 1  −1 dx 1  x 2 dy 2  0 (4 + z)dx. 12.2. T´ıch phˆan 3-l´o . p 137 Gia ’ i. Ta t´ınh liˆen tiˆe ´ p ba t´ıch phˆan x´ac di . nh thˆong thu . `o . ng b˘a ´ t d ˆa ` ut`u . t´ıch phˆan trong I(x, y)= 2  0 (4 + z)dz =4z   2 0 + z 2 2    2 0 = 10; I(x)= 1  x 2 I(x, y)dy =10 1  x 2 dy = 10(1 − x 2 ); I = 1  −1 I(x)dx = 1  −1 10(1 − x 2 )dx = 40 3 ·  V´ı du . 2. T´ınh t´ıch phˆan I =  D (x + y + z)dxdydz, trong d ´omiˆe ` n D du . o . . c gi´o . iha . nbo . ’ i c´ac m˘a . t ph˘a ’ ng to . ad ˆo . v`a m˘a . t ph˘a ’ ng x + y + z =1. Gia ’ i. Miˆe ` n D d ˜a cho l`a mˆo . tt´u . diˆe . nc´oh`ınh chiˆe ´ u vuˆong g´oc trˆen m˘a . t ph˘a ’ ng Oxy l`a tam gi´ac gi´o . iha . nbo . ’ i c´ac d u . `o . ng th˘a ’ ng x =0, y =0,x + y = 1. R˜o r`ang l`a x biˆe ´ n thiˆen t`u . 0d ˆe ´ n1(doa . n[0, 1] l`a h`ınh chiˆe ´ ucu ’ a D lˆen tru . c Ox). Khi cˆo ´ d i . nh x,0 x  1th`ıy biˆe ´ n thiˆen t`u . 0d ˆe ´ n1−x.Nˆe ´ ucˆo ´ di . nh ca ’ x v`a y (0  x  1, 0  y  1 −x) th`ı d iˆe ’ m(x, y, z)biˆe ´ n thiˆen theo du . `o . ng th˘a ’ ng d ´u . ng t`u . m˘a . t ph˘a ’ ng z =0d ˆe ´ nm˘a . t ph˘a ’ ng x + y + z =1,t´u . cl`az biˆe ´ n thiˆen t`u . 0d ˆe ´ n 1 − x −y. Theo cˆong th´u . c (12.16) ta c´o I = 1  0 dx 1−x  0 dy 1−x−y  0 (x + y + z)dz. 138 Chu . o . ng 12. T´ıch phˆan h`am nhiˆe ` ubiˆe ´ n Dˆe ˜ d`ang thˆa ´ yr˘a ` ng I = 1  0 dx 1−x  0  xz + yz + z 2 2     1−x−y 0 dy = 1 2 1  0  y − yx 2 − xy 2 − y 3 3     1−x 0  dx = 1 6 1  0 (2 − 3x + x 3 )dx = 1 8 ·  V´ı d u . 3. T´ınh I =  D dxdydz (x + y + z) 3 , trong d´omiˆe ` n D du . o . . c gi´o . i ha . nbo . ’ i c´ac m˘a . t ph˘a ’ ng x + z =3,y =2,x =0,y =0,z =0. Gia ’ i. Miˆe ` n D d ˜a cho l`a mˆo . th`ınh l˘ang tru . c´o h`ınh chiˆe ´ u vuˆong g´oc lˆen m˘a . t ph˘a ’ ng Oxy l`a h`ınh ch˜u . nhˆa . t D(x, y)=  (x, y):0 x  3, 0  y  2  .V´o . id iˆe ’ m M(x, y)cˆo ´ di . nh thuˆo . c D(x, y)diˆe ’ m (x, y, z) ∈ D biˆe ´ n thiˆen trˆen d u . `o . ng th˘a ’ ng d ´u . ng t`u . m˘a . t ph˘a ’ ng Oxy (z =0)d ˆe ´ nm˘a . t ph˘a ’ ng x + z =3,t´u . cl`az biˆe ´ n thiˆen t`u . 0d ˆe ´ n3−x: 0  z  3 − x.T`u . d ´o theo (12.17) ta c´o  D f(M)dxdydz =  D(x,y) dxdy z=3−x  z=0 (x + y + z +1) −3 dz =  D(x,y)  (x + y + z +1) −2 −2    3−x 0  dxdy = ···= 4ln2− 1 8 ·  V´ı du . 4. T´ınh t´ıch phˆan  D (x 2 + y 2 + z 2 )dxdydz, trong d´omiˆe ` n D d u . o . . c gi´o . iha . nbo . ’ im˘a . t3(x 2 + y 2 )+z 2 =3a 2 . Gia ’ i. Phu . o . ng tr`ınh m˘a . tbiˆen cu ’ a D c´o thˆe ’ viˆe ´ tdu . ´o . ida . ng x 2 a 2 + y 2 b 2 + z 2 (a √ 3) 2 =1. 12.2. T´ıch phˆan 3-l´o . p 139 D´o l`a m˘a . t elipxoid tr`on xoay, t´u . cl`aD l`a h`ınh elipxoid tr`on xoay. H`ınh chiˆe ´ u vuˆong g´oc D(x, y)cu ’ a D lˆen m˘a . t ph˘a ’ ng Oxy l`a h`ınh tr`on x 2 + y 2  a 2 .Dod´o ´ap du . ng c´ach lˆa . p luˆa . nnhu . trong c´ac v´ıdu . 2 v`a 3 ta thˆa ´ yr˘a ` ng khi d iˆe ’ m M(x, y) ∈ D(x, y)du . o . . ccˆo ´ d i . nh th`ı diˆe ’ m (x, y, z)cu ’ amiˆe ` n D biˆe ´ n thiˆen trˆen d u . `o . ng th˘a ’ ng d ´u . ng M(x,y)t`u . m˘a . tbiˆen du . ´o . icu ’ a D z = −  3(a 2 − x 2 − y 2 ) d ˆe ´ nm˘a . tbiˆen trˆen z =+  3(a 2 − x 2 − y 2 ). T`u . d ´o theo (12.17) ta c´o I =  D(x,y) dxdy + √ 3(a 2 −x 2 −y 2 )  − √ 3(a 2 −x 2 −y 2 ) (x 2 + y 2 + z 2 )dz =2a 2 √ 3  x 2 +y 2 a 2  a 2 − x 2 − y 2 dxdy = |chuyˆe ’ n sang to . adˆo . cu . . c| =2a 2 √ 3  ra √ a 2 −r 2 rdrdϕ = a 2 √ 3 2π  0 dϕ a  0 (a 2 − r 2 ) 1/2 rdr = 4πa 5 √ 3 ·  V´ı d u . 5. T´ınh thˆe ’ t´ıch cu ’ avˆa . tthˆe ’ gi´o . iha . nbo . ’ i c´ac m˘a . t ph˘a ’ ng x + y + z =4,x =3,y =2,x =0,y =0,z =0. Gia ’ i. Miˆe ` n D d ˜a cho l`a mˆo . th`ınh lu . cdiˆe . n trong khˆong gian. N´o c´o h`ınh chiˆe ´ u vuˆong g´oc D(x, y) lˆen m˘a . t ph˘a ’ ng Oxy l`a h`ınh thang vuˆong gi´o . iha . nbo . ’ i c´ac d u . `o . ng th˘a ’ ng x =0,y =0,x =3,y =2v`a 140 Chu . o . ng 12. T´ıch phˆan h`am nhiˆe ` ubiˆe ´ n x + y = 4. Do d´o´apdu . ng (12.17) ta c´o V D =  D dxdydz =  D(x,y) dxdy 4−x−y  0 dz =  D(x,y) (4 − x − y)dxdy = 1  0 dy 3  0 (4 − x − y)dx + 2  1 dy 4−y  0 (4 − x −y)dx = 1  0  (4 − y)x − x 2 2     3 0  dy + 2  1  (4 − y)x − x 2 2     4−y 0  dy = 1  0  15 2 −3y  dy + 1 2 2  1 (4 − y) 2 dy = 55 6 ·  V´ı du . 6. T´ınh t´ıch phˆan I =  D z  x 2 + y 2 dxdydz, trong d ´omiˆe ` n D gi´o . iha . nbo . ’ im˘a . t ph˘a ’ ng y =0,z =0,z = a v`a m˘a . t tru . x 2 + y 2 =2x (x  0, y  0, a>0). Gia ’ i. Chuyˆe ’ n sang to . ad ˆo . tru . ta thˆa ´ yphu . o . ng tr`ınh m˘a . t tru . x 2 + y 2 =2x trong to . adˆo . tru . c´o da . ng r = 2 cos ϕ,0 ϕ  π 2 (h˜ay v˜e h`ınh !). Do d ´o theo cˆong th´u . c (12.20) ta c´o I = π/2  0 dϕ 2 cosϕ  0 r 2 dr a  0 zdz = a 2 2 π/2  0 dϕ 2 cosϕ  0 r 2 dr = 4a 2 3 π/2  0 cos 3 ϕdϕ = 8 9 a 2 .  V´ı du . 7. T´ınh t´ıch phˆan I =  D (x 2 + y 2 )dxdydz, 12.2. T´ıch phˆan 3-l´o . p 141 nˆe ´ umiˆe ` n D l`a nu . ’ a trˆen cu ’ a h`ınh cˆa ` u x 2 + y 2 + z 2  R 2 , z  0. Gia ’ i. Chuyˆe ’ n sang to . ad ˆo . cˆa ` u, miˆe ` nbiˆe ´ n thiˆen D ∗ cu ’ a c´ac to . adˆo . cˆa ` utu . o . ng ´u . ng khi d iˆe ’ m(x, y, z)biˆe ´ n thiˆen trong D l`a c´o da . ng D ∗ :0 ϕ<2π, 0  θ  π 2 , 0  r  R. T`u . d ´o I =  D ∗ r 2 sin 2 θ · r 2 sin θdrdϕdθ = 2π  0 dϕ π/2  0 sin 3 θdθ R  0 r 4 dr = 4 15 πR 5 .  B ` AI T ˆ A . P T´ınh c´ac t´ıch phˆan l˘a . p sau 1. 1  0 dx √ x  0 ydy 2−2x  1−x dz.(DS. 1 12 ) 2. a  0 ydy h  0 dx a−y  0 dz.(DS. a 3 h 6 ) 3. 2  0 dy 2  √ 2y−y 2 xdx 3  0 z 2 dz.(DS. 30) 4. 1  0 dx 1−x  0 dy 1−x−y  0 dz (1 + x + y + z) 3 .(DS. ln 2 2 − 5 16 ) 5. c  0 dz b  0 dy a  0 (x 2 + y 2 + z 2 )dx.(DS.  abc 3 (a 2 + b 2 + c 2 )  ) [...]... = 0 a b c 29 ax = y + z , x = a (DS abc ) 6 πa3 ) (DS 2 30 2z = x2 + y 2, z = 2 (DS 12) (DS 4π) 2 2 31 z = x2 + y 2, x2 + y 2 + z 2 = 2 (DS 32 z = x2 + y 2 , z = x2 + y 2 33 x2 + y 2 − z = 1, z = 0 34 2z = x2 + y 2, y + z = 4 35 x2 y 2 z 2 + + 2 = 1 a2 b2 c (DS (DS π √ [8 2 − 7]) 6 π ) 6 π ) 2 81π ) (DS 4 (DS 4 πabc) 3 12 .3 T´ phˆn du.`.ng ıch a o 12 .3. 1 ’ C´c dinh ngh˜ co ban a ıa ’ ’ ’ a a e ... th`nh t´ phˆn 2-l´.p T` phu.o.ng tr`nh cua (σ) r´t ıch a a a ıch a o u ı u 1 u o ra z = (6 − x − 2y) T` d´ 3 √ 14 2 2 dxdy dS = 1 + zx + zy dxdy = 2 Do d´ o √ 14 I= 3 3 [(6x + 4y + (6 − x − 2y)]dxdy 3 ∆OAB √ 14 = 3 6−2y 3 dy 0 √ 14 = 3 (5x + 2y + 6)dx 0 3 5 2 x + 2xy + 6x 2 0 6−2y 0 √ dy = 54 14 ´ ` ıch a a e e Chu.o.ng 12 T´ phˆn h`m nhiˆu biˆn 164 1 + 4x2 + 4y 2 dS, (σ) l` phˆn paraboloid tr`n... · 2x √ √ 1 √ 1 + 2x √ dx = [5 5 − 3 3] 6 2x 1 o V´ du 2 T´ dˆ d`i cua du.`.ng astroid x = a cos3 t, y = a sin3 t, ı ınh o a ’ t ∈ [0, 2π] ’ Giai Ta ´p dung cˆng th´.c: dˆ d`i (L) = ds Trong tru.`.ng a o u o o a L a o ho.p n`y ta c´ x = −3a cos2 t sin t, y = 3a sin2 t cos t, ds = 3a sin 2tdt 2 ´ V` du.`.ng cong dˆi x´.ng v´.i c´c truc toa dˆ nˆn ı o o a o u o e π/2 3a − cos 2t sin 2tdt = 6a 2... a e e a a ı 1 c tiˆp (DS ´ ) e tru 30 ex [(1 − cos y)dx − (y − sin y)dy], C l` biˆn cua tam gi´c ABC a e ’ a 33 C a v´.i A = (1, 1), B = (0, 2) v` C = (0, 0) (DS 2(2 − e)) o o 12 .3 T´ phˆn d u.`.ng ıch a 34 157 (xy + x + y)dx + (xy + x − y)dy, trong d´ C l` o a C x2 y 2 a) elip 2 + 2 = 1; a b 3 `.ng tr`n x2 + y 2 = ax (a > 0) (DS a) 0; b) − πa ) b) du o o 8 πR4 35 xy 2dx − x2ydy, C l` du.`.ng tr`n... BC ta c´ x + y = 1 ⇒ y = −x + 1, dy = −dx Do d´ e o 0 5 [3x2 + (1 − x) − x + 2(1 − x2)]dx = − · 3 = BC 1 o 12 .3 T´ phˆn d u.`.ng ıch a 151 c) Trˆn canh CA ta c´ x = 0 ⇒ dx = 0 v` do d´ e o a o 0 2y 2dy = =− 2 · 3 1 CA Nhu vˆy a =1− 5 2 + = 0 3 3 L (x +y)dx −(x −y)dy, trong d´ L l` du.`.ng o a o V´ du 5 T´ t´ phˆn ı ınh ıch a L x2 y 2 o o elip 2 + 2 = 1 c´ dinh hu.´.ng du.o.ng a b + ’ ´ ` ’ e ı... Green ta c´ o u = L [(4y + 3) − 4y]dxdy = 3 ∆ABC dxdy ∆ABC = 3S∆ABC = 3 ` ˆ BAI TAP T´ c´c t´ phˆn du.`.ng theo dˆ d`i sau dˆy ınh a ıch a o o a a √ ’ ´ (x + y)ds, C l` doa n th˘ng nˆi A(9, 6) v´.i B(1, 2) (DS 36 5) a a o o 1 C 2 xyds, C l` biˆn h` vuˆng |x| + |y| = a, a > 0 (DS 0) a e ınh o C (x + y)ds, C l` biˆn cua tam gi´c dınh A(1, 0), B(0, 1), C(0, 0) a e ’ a ’ 3 C (DS 1 + √ 2) √ ds ’ ´ ,... dinh ngh˜a ıa a ıch a a ı o i ’ bo n−1 def i P (Ni )m(σxy ) d→0 (σ) (12 . 35 ) i Q(Ni )m(σxz ) (12 .36 ) i R(Ni )m(σyz ) P (M)dxdy = lim (12 .37 ) i=0 n−1 def Q(M)dxdz = lim d→0 (σ) i=0 n−1 def R(M )dydz = lim d→0 (σ) i=0 ´ ` ıch a a e e Chu.o.ng 12 T´ phˆn h`m nhiˆu biˆn 160 ´ ´ ` u o o nˆu c´c gi´.i han o vˆ phai (12 . 35 )-(12 .37 ) tˆn tai h˜.u han khˆng phu e a o ’ e ’ ’ e thuˆc v`o ph´p phˆn hoach... a 2 + y2 + 4 x C (DS √ 5 +3 ) ln 4 o (x2 + y 2 + z 2 )ds, C l` cung du.`.ng cong x = a cos t, y = a sin t, a 10 C z = bt; 0 t 2π, a > 0, b > 0 2π √ 2 a + b2 (3a2 + 4π 2b2 )) (DS 3 o x2ds, C l` du.`.ng tr`n a o 11 C  x2 + y 2 + z 2 = a2 x + y + z = 0 ˜ ` ’ a ’ a Chı dˆ n Ch´.ng to r˘ng u x2 ds = C ra I= (DS 1 3 z 2 ds v` t` d´ suy a u o y 2ds = C (x2 + y 2 + z 2)ds C 2πa3 ) 3 C ´ ` ıch a a e e Chu.o.ng... (DS 5 ln 2) a a o o x−y 4 C o x2 + y 2ds, C l` du.`.ng tr`n x2 + y 2 = ax a o 5 (DS 2a2 ) C o (x2 + y 2)n ds, C l` du.`.ng tr`n x2 + y 2 = a2 a o 6 C √ 7 e C x2 +y 2 ds, C l` biˆn h`nh quat tr`n a e ı o (DS 2πa2n+1) o 12 .3 T´ phˆn d u.`.ng ıch a (r, ϕ) : 0 r 1 53 a, 0 π 4 ϕ (DS 2(ea − 1) + πaea ) 4 ` ` ` a o a xyds, C l` mˆt phˆn tu elip n˘m trong g´c phˆn tu I a o a 8 C ab a2 + ab + b2 · ) 3 a+b... Chu.o.ng 12 T´ phˆn h`m nhiˆu biˆn 158 (x + y)2dx − (x2 + y 2)dy, C l` biˆn cua ∆ABC v´.i dınh a e ’ o ’ 40 C 2 A(1, 1), B (3, 2) v` C(2, 5) (DS −46 ) a 3 (y − x2)dx + (x + y 2)dy, C l` biˆn h`nh quat b´n k´ R v` a e ı a a ınh 41 C g´c ϕ (0 o ϕ π ) (DS 0) 2 y 2 dx + (x + y)2dy, C l` biˆn cua h`nh tam gi´c ∆ABC v´.i a e ’ ı a o 42 C A(a, 0), B(a, a), C(0, a) (DS 2a3 ) 3 12.4 T´ phˆn m˘t ıch a a 12.4.1 . (D S. 1 6 ) 54 . x =0,y =0,z =0,x + y =1,z = x 2 + y 2 .(DS. 1 6 ) 55 . z = x 2 + y 2 , y = x 2 , y =1,z = 0. (DS. 88 1 05 ) 56 . z =  x 2 + y 2 , x 2 + y 2 = a 2 , z = 0. (DS. 2 3 πa 3 ) 57 . z = x 2 +. 7]) 32 . z =  x 2 + y 2 , z = x 2 + y 2 .(DS. π 6 ) 33 . x 2 + y 2 −z =1,z = 0. (DS. π 2 ) 34 . 2z = x 2 + y 2 , y + z = 4. (DS. 81π 4 ) 35 . x 2 a 2 + y 2 b 2 + z 2 c 2 = 1. (DS. 4 3 πabc) 12 .3 T´ıch. z 2 )dz =2a 2 √ 3  x 2 +y 2 a 2  a 2 − x 2 − y 2 dxdy = |chuyˆe ’ n sang to . adˆo . cu . . c| =2a 2 √ 3  ra √ a 2 −r 2 rdrdϕ = a 2 √ 3 2π  0 dϕ a  0 (a 2 − r 2 ) 1/2 rdr = 4πa 5 √ 3 ·  V´ı d u . 5.