12.4. T´ıch phˆan m˘a . t 165 V´ı du . 4. T´ınh t´ıch phˆan  (σ) 2dxdy + ydxdz −x 2 zdydz, trong d´o(σ) l`a ph´ıa trˆen cu ’ a phˆa ` n elipxoid 4x 2 + y 2 +4z 2 =1n˘a ` m trong g´oc phˆa ` n t´am I. Gia ’ i. Ta viˆe ´ tt´ıch phˆan d ˜a cho du . ´o . ida . ng I =2  (σ) dxdy +  (σ) ydydz −  (σ) x 2 zdydz. v`a su . ’ du . ng phu . o . ng tr`ınh cu ’ am˘a . t(σ)d ˆe ’ biˆe ´ ndˆo ’ imˆo ˜ i t´ıch phˆan. Lu . u ´yr˘a ` ng cos α>0, cos β>0, cos γ>0. (i) V`ı h`ınh chiˆe ´ ucu ’ am˘a . t(σ)lˆen m˘a . t ph˘a ’ ng Oxy l`a phˆa ` ntu . h`ınh elip x 2 1 2 + y 2 2 2  1nˆen I 1 =  (σ) dxdy =  D(x,y) dxdy = π 2 (v`ı diˆe . n t´ıch elip = 2π) (ii) H`ınh chiˆe ´ ucu ’ a(σ)lˆen m˘a . t ph˘a ’ ng Oxz l`a phˆa ` ntu . h`ınh tr`on 4x 2 +4z 2  4 ⇔ x 2 + z 2  1. M˘a . t kh´ac t `u . phu . o . ng tr`ınh m˘a . tr´ut ra y =2  1 − x 2 − y 2 v`a do d´o I 2 =  (σ) ydxdz =2  D(x,y) √ 1 − x 2 − z 2 dxdz = |chuyˆe ’ n sang to . adˆo . cu . . c| =2 π/2  0 dϕ 1  0 √ 1 −r 2 rdr = π 3 · (iii) H`ınh chiˆe ´ ucu ’ a(σ)lˆen m˘a . t ph˘a ’ ng Oyz l`a mˆo . t phˆa ` ntu . h`ınh elip y 2 4 + z 2  1(y  0, z  0). T`u . phu . o . ng tr`ınh m˘a . t(σ)r´ut ra 166 Chu . o . ng 12. T´ıch phˆan h`am nhiˆe ` ubiˆe ´ n x =  1 − y 2 4 − z 2 rˆo ` ithˆe ´ v`ao h`am du . ´o . idˆa ´ u t´ıch phˆan cu ’ a I 3 : I 3 =  (σ) x 2 zdydz =  D(y,z) z  1 − y 2 4 − z 2  dydz = 1  0 dz 2 √ 1−z 2  0 z  1 − y 2 4 −z 2  dy = ···= 4 15 · Nhu . vˆa . y I =2I 1 + I 2 − I 3 = 4π 3 − 4 15 ·  V´ı d u . 5. T´ınh  (σ) − ydydz, trong d´o(σ) l`a m˘a . tcu ’ at´u . diˆe . n gi´o . iha . n bo . ’ im˘a . t ph˘a ’ ng x +y +z = 1 v`a c´ac m˘a . t ph˘a ’ ng to . ad ˆo . , t´ıch phˆan du . o . . c lˆa ´ y theo ph´ıa trong cu ’ at´u . diˆe . n. Gia ’ i. M˘a . t ph˘a ’ ng x + y + z =1c˘a ´ t c´ac tru . cto . ad ˆo . ta . i A(1, 0, 0), B(0, 1,0) v`a C =(0, 0, 1). Ta k´y hiˆe . ugˆo ´ cto . ad ˆo . l`a O(0, 0,0). T`u . d ´o suy ra m˘a . tk´ın (σ)gˆo ` mt`u . 4 h`ınh tam gi´ac ∆ABC,∆BCO,∆ACO v`a ∆ABO. Do vˆa . y t´ıch phˆan d ˜a cho l`a tˆo ’ ng cu ’ abˆo ´ n t´ıch phˆan. (i) T´ıch phˆan I 1 =  ABC ydxdz.R´ut y t`u . phu . o . ng tr`ınh m˘a . t(σ) ⊃ ∆ABC ta c´o y =1−x −z v`a do d ´o I 1 = −  ACO (1 − x −z)dxdz = 1  0 dx 1−x  0 (x + z −1)dz = − 1 6 · (Lu . u´yr˘a ` ng cos β = cos(n , O y ) < 0 v`ı vecto . n lˆa . pv´o . ihu . ´o . ng du . o . ng tru . c Oy mˆo . t g´oc t`u, do d ´o tru . ´o . c t´ıch phˆan theo ∆ACO xuˆa ´ thiˆe . ndˆa ´ u tr `u . ) (ii)  (BCD) ydxdz =  (ABO) ydxdz =0 12.4. T´ıch phˆan m˘a . t 167 v`ım˘a . t ph˘a ’ ng BCO v`a ABO dˆe ` u vuˆong g´oc v´o . im˘a . t ph˘a ’ ng Oxz. (iii)  (ACO) ydxdz =  ACO 0dxdz =0. Vˆa . y I = − 1 6 .  V´ı du . 6. T´ınh t´ıch phˆan I =  (σ) x 3 dydz + y 3 dzdx + z 3 dxdy, trong d ´o ( σ) l`a ph´ıa ngo`ai m˘a . tcˆa ` u x 2 + y 2 + z 2 = R 2 . Gia ’ i. ´ Ap du . ng cˆong th´u . c Gauss-Ostrogradski ta c´o  (σ) =3  D (x 2 + y 2 + z 2 )dxdydz trong d ´o D ⊂ R 3 l`a miˆe ` nv´o . i biˆen l`a m˘a . t(σ). Chuyˆe ’ n sang to . ad ˆo . cˆa ` u ta c´o 3  D (x 2 + y 2 + z 2 )dxdydz =3 2π  0 dϕ π  0 sin θdθ R  0 r 4 dr = 12πR 5 5 · Vˆa . y I = 12πR 5 5 ·  V´ı du . 7. T´ınh t´ıch phˆan  L x 2 y 3 dx + dy + zdz, trong d´o L l`a du . `o . ng tr`on x 2 + y 2 =1,z = 0, c`on m˘a . t(σ) l`a ph´ıa ngo`ai cu ’ anu . ’ am˘a . tcˆa ` u x 2 + y 2 + z 2 =1,z>0v`aL c´o di . nh hu . ´o . ng du . o . ng. Gia ’ i. Trong tru . `o . ng ho . . p n`ay P = x 2 y 3 , Q =1,R = z.Dod´o ∂Q ∂x − ∂P ∂y = −3x 2 y 2 , ∂R ∂y − ∂Q ∂z =0, ∂P ∂z − ∂R ∂x =0 168 Chu . o . ng 12. T´ıch phˆan h`am nhiˆe ` ubiˆe ´ n v`a do d´o theo cˆong th´u . c Stokes ta c´o  L = −3  (σ) x 2 y 2 dxdy = − π 8 ·  B ` AI T ˆ A . P T´ınh c´ac t´ıch phˆan m˘a . t theo diˆe . n t´ıch sau d ˆay 1.  (Σ) (x + y + z)dS, (Σ) l`a m˘a . tlˆa . pphu . o . ng 0  x  1, 0  1, 0  z  1. (D S. 9) 2.  (Σ) (2x+y +z)dS, (Σ) l`a phˆa ` nm˘a . t ph˘a ’ ng x+y +z =1n˘a ` m trong g´oc phˆa ` n t´am I.(D S. 2 √ 3 3 ) 3.  (Σ)  z +2x + 4y 3  dS, (Σ) l`a phˆa ` nm˘a . t ph˘a ’ ng 6x +4y +3z =12 n˘a ` m trong g´oc phˆa ` n t´am I. (D S. 4 √ 61) 4.  (σ)  x 2 + y 2 dS, (Σ) l`a phˆa ` nm˘a . t n´on z 2 = x 2 + y 2 ,0 z  1. (D S. 2 √ 2π 3 ) 5.  (Σ) (y + z + √ a 2 − x 2 )dS, (Σ) l`a phˆa ` nm˘a . t tru . x 2 + y 2 = a 2 n˘a ` m gi˜u . a hai m˘a . t ph˘a ’ ng z =0v`az = h.(D S. ah(4a + πh)) 6.  (Σ)  y 2 − x 2 dS, (Σ) l`a phˆa ` nm˘a . t n´on z 2 = x 2 + y 2 n˘a ` m trong m˘a . t tru . x 2 + y 2 = a 2 .(DS. 8a 3 3 ) 12.4. T´ıch phˆan m˘a . t 169 7.  (Σ) (x + y + z)dS, (Σ) l`a nu . ’ a trˆen cu ’ am˘a . tcˆa ` u x 2 + y 2 + z 2 = a 2 . (D S. πa 3 ) 8.  (Σ)  x 2 + y 2 dS, (Σ) l`a m˘a . tcˆa ` u x 2 + y 2 + z 2 = a 2 .(DS. 8πa 3 3 ) 9.  (Σ) dS (1 + x + y) , (Σ) l`a biˆen cu ’ at´u . diˆe . n x´ac d i . nh bo . ’ ibˆa ´ tphu . o . ng tr`ınh x+y+z  1, x  0, y  0, z  0. (D S. 1 3 (3− √ 3) +( √ 3−1) ln 2) 10.  (Σ) (x 2 + y 2 )dS, (Σ) l`a phˆa ` nm˘a . t paraboloid x 2 + y 2 =2z du . o . . c c˘a ´ trabo . ’ im˘a . t ph˘a ’ ng z = 1. (D S. 55 + 9 √ 3 65 ) 11.  (Σ)  1+4x 2 +4y 2 dS, (Σ) l`a phˆa ` nm˘a . t paraboloid z =1−x 2 −y 2 gi´o . iha . nbo . ’ i c´ac m˘a . t ph˘a ’ ng z =0v`az = 1. (D S. 3π) 12.  (Σ) (x 2 + y 2 )dS, (Σ) l`a phˆa ` nm˘a . t n´on z =  x 2 + y 2 n˘a ` mgi˜u . a c´ac m˘a . t ph˘a ’ ng z =0v`az = 1. (D S. π √ 2 2 ) 13.  (Σ) (xy + yz + zx)dS, (Σ) l`a phˆa ` nm˘a . t n´on z =  x 2 + y 2 n˘a ` m trong m˘a . t tru . x 2 + y 2 =2ax (a>0). (DS. 64a 4 √ 2 15 ) 14.  (Σ) (x 2 + y 2 + z 2 )dS, (Σ) l`a ma . tcˆa ` u. (DS. 4π) 15.  (Σ) xds, (Σ) l`a phˆa ` nm˘a . tdu . o . . cc˘a ´ trat`u . partab oloid 10x = y 2 +z 2 170 Chu . o . ng 12. T´ıch phˆan h`am nhiˆe ` ubiˆe ´ n bo . ’ im˘a . t ph˘a ’ ng x = 10. (D S. 50π 3 (1 + 25 √ 5)) Su . ’ du . ng cˆong th´u . c t´ınh diˆe . n t´ıch m˘a . t S(Σ) =  (Σ) dS dˆe ’ t´ınh diˆe . n t´ıch cu ’ a phˆa ` nm˘a . t (Σ) nˆe ´ u 16. (Σ) l`a phˆa ` nm˘a . t ph˘a ’ ng 2x +2y + z =8a n˘a ` m trong m˘a . t tru . x 2 + y 2 = R 2 .(DS. 3πR 2 ) 17. (Σ) l`a phˆa ` nm˘a . t tru . y + z 2 = R 2 n˘a ` m trong m˘a . t tru . x 2 + y 2 = R 2 .(DS. 8R 2 ) 18. (Σ) l`a phˆa ` nm˘a . t paraboloid x 2 + y 2 =6z n˘a ` m trong m˘a . t tru . x 2 + y 2 = 27. (DS. 42π) 19. (Σ) l`a phˆa ` nm˘a . tcˆa ` u x 2 + y 2 + z 2 =3a 2 n˘a ` m trong paraboloid x 2 + y 2 =2az.(DS. 2πa 2 (3 − √ 3)) 20. (Σ) l`a phˆa ` nm˘a . t n´on z 2 =2xy n˘a ` m trong g´oc phˆa ` n t´am I gi˜u . a hai m˘a . t ph˘a ’ ng x =2,y = 4. (D S. 16) 21. (Σ) l`a phˆa ` nm˘a . t tru . x 2 + y 2 = Rx n˘a ` m trong m˘a . tcˆa ` u x 2 + y 2 + z 2 = R 2 .(DS. 4R 2 ) T´ınh c´ac t´ıch phˆan m˘a . t theo to . ad ˆo . sau: 22.  (Σ) dxdy, (Σ) l`a ph´ıa ngo`ai phˆa ` nm˘a . t n´on z =  x 2 + y 2 khi 0  z  1. (D S. −π) 23.  (Σ) ydzdx, (Σ) l`a ph´ıa trˆen cu ’ a phˆa ` nm˘a . t ph˘a ’ ng x + y + z = a (a>0) n˘a ` m trong g´oc phˆa ` n t´am I.(D S. a 3 6 ) 24.  (Σ) xdydz + ydzdx+ zdxdy, (Σ) l`a ph´ıa trˆen cu ’ a phˆa ` nm˘a . t ph˘a ’ ng x + z −1=0n˘a ` mgi˜u . a hai m˘a . t ph˘a ’ ng y =0v`ay = 4 v`a thuˆo . c v`ao g´oc phˆa ` n t´am I. (D S. 4) 12.4. T´ıch phˆan m˘a . t 171 25.  (Σ) − xdydz + zdzdx +5dxdy, (Σ) l`a ph´ıa trˆen cu ’ a phˆa ` nm˘a . t ph˘a ’ ng 2x +3y + z = 6 thuˆo . c g´oc phˆa ` n t´am I. (D S. 6) 26.  (Σ) yzdydz + xzdxdz + xydxdy, (Σ) l`a ph´ıa trˆen cu ’ a tam gi´ac ta . o bo . ’ i giao tuyˆe ´ ncu ’ am˘a . t ph˘a ’ ng x + y + z = a v´o . i c´ac m˘a . t ph˘a ’ ng to . a d ˆo . .(DS. a 4 8 ) 27.  (Σ) x 2 dydz + z 2 dxdy, (Σ) l`a ph´ıa ngo`ai cu ’ a phˆa ` nm˘a . t n´on x 2 + y 2 = z 2 ,0 z  1. (DS. − 4 3 ) 28.  (Σ) xdydz + ydzdx + zdxdy, (Σ) l`a ph´ıa ngo`ai phˆa ` nm˘a . tcˆa ` u x 2 + y 2 + z 2 = a 2 .(DS. 4πa 3 ) 29.  (σ) x 2 dydz −y 2 dzdx + z 2 dxdy, (Σ) l`a ph´ıa ngo`ai cu ’ am˘a . tcˆa ` u x 2 + y 2 + z 2 = R 2 thuˆo . c g´oc phˆa ` n t´am I. (DS. πa 4 8 ) 30.  (Σ) 2dxdy + ydzdx − x 2 zdydz, (Σ) l`a ph´ıa ngo`ai cu ’ a phˆa ` nm˘a . t elipxoid 4x 2 + y 2 +4z 2 = 4 thuˆo . c g´oc phˆa ` n t´am I. (DS. 4π 3 − 4 15 ) 31.  (Σ) (y 2 + z 2 )dxdy, (Σ) l`a ph´ıa ngo`ai cu ’ am˘a . t tru . z 2 =1− x 2 , 0  y  1. (D S. π 3 ) 32.  (Σ) (z −R) 2 dxdy, (Σ) l`a ph´ıa ngo`ai cu ’ anu . ’ am˘a . tcˆa ` u x 2 + y 2 +(z −R) 2 = R 2 , R  z  2R.(DS. − 5π 24 ) 172 Chu . o . ng 12. T´ıch phˆan h`am nhiˆe ` ubiˆe ´ n 33.  (Σ) x 2 dydz + y 2 dzdx + z 2 dxdy, (Σ) l`a ph´ıa ngo`ai cu ’ a phˆa ` nm˘a . t cˆa ` u x 2 + y 2 + z 2 = a 2 thuˆo . c g´oc phˆa ` n t´am I. (DS. 3πa 4 8 ) 34.  (Σ) z 2 dxdy,(σ) l`a ph´ıa trong cu ’ am˘a . t elipxoid x 2 + y 2 +2z 2 = 2. (DS. 0) 35.  (Σ) (z +1)dxdy, (Σ) l`a ph´ıa ngo`ai cu ’ am˘a . tcˆa ` u x 2 + y 2 + z 2 = R 2 .(DS. 4πR 3 3 ) 36.  (Σ) x 2 dydz + y 2 dzdx + z 2 dxdy, (Σ) l`a ph´ıa ngo`ai cu ’ am˘a . tcˆa ` u (x −a) 2 +(y − b) 2 +(z − c) 2 = R 2 .(DS. 8πR 3 3 (a + b + c)) 37.  (Σ) x 2 y 2 zdxdy, (Σ) l`a ph´ıa trong cu ’ anu . ’ adu . ´o . im˘a . tcˆa ` u x 2 + y 2 + z 2 = R 2 .(DS. 2πR 7 105 ) 38.  (Σ) xzdxdy + xydydz + yzdxdz, (Σ) l`a ph´ıa ngo`ai cu ’ at´u . diˆe . nta . o bo . ’ i c´ac m˘a . t ph˘a ’ ng to . ad ˆo . v`a m˘a . t ph˘a ’ ng x + y + z = 1. (DS. 1 8 ) Chı ’ dˆa ˜ n. Su . ’ du . ng nhˆa . n x´et nˆeu trong phˆa ` nl´ythuyˆe ´ t. 39.  (Σ) yzdydz + xzdxdz + xydxdy, (Σ) l`a ph´ıa ngo`ai cu ’ am˘a . tbiˆen t´u . diˆe . nlˆa . pbo . ’ i c´ac m˘a . t ph˘a ’ ng x =0,y =0,z =0,x + y + z = a. (D S. 0) 40.  (Σ) x 2 dydz + y 2 dzdx + z 2 dxdy, (Σ) l`a ph´ıa ngo`ai cu ’ anu . ’ a trˆen 12.4. T´ıch phˆan m˘a . t 173 m˘a . tcˆa ` u x 2 + y 2 + z 2 = R 2 (z  0). (DS. πR 4 2 ) ´ Ap du . ng cˆong th´u . c Gauss-Ostrogradski d ˆe ’ t´ınh t´ıch phˆan m˘a . t theo ph´ıa ngo`ai cu ’ am˘a . t (Σ) (nˆe ´ um˘a . t khˆong k´ın th`ı bˆo ’ sung d ˆe ’ n´o tro . ’ th`anh k´ın) 41.  (Σ) x 2 dydz + y 2 dzdx + z 2 dxdy, (Σ) l`a m˘a . tcˆa ` u (x −a) 2 +(y −b) 2 +(z −c) 2 = R 2 .(DS. 8π 3 (a + b + c)R 3 ) 42.  (Σ) xdydz + ydzdx + zdxdy, (Σ) l`a m˘a . tcˆa ` u x 2 + y 2 + z 2 = R 2 . (D S. 4πR 3 ) 43.  (Σ) 4x 3 dydz +4y 3 dzdx − 6z 2 dxdy, (Σ) l`a biˆen cu ’ a phˆa ` n h`ınh tru . x 2 + y 2  a 2 ,0 z  h.(DS. 6πa 2 (a 2 − h 2 )) 44.  (σ) (y −z)dydz +(z −x)dzdx +(x − y)dxdy, (Σ) l`a phˆa ` nm˘a . t n´on x 2 + y 2 = z 2 ,0 x  h.(DS. 0) Chı ’ dˆa ˜ n. V`ı (Σ) khˆong k´ın nˆen cˆa ` nbˆo ’ sung phˆa ` nm˘a . t ph˘a ’ ng z = h n˘a ` m trong n´on d ˆe ’ thu du . o . . cm˘a . tk´ın. 45.  (Σ) dydz + zxdzdx + xydxdy, (Σ) l`a biˆen cu ’ amiˆe ` n {(x, y, z):x 2 + y 2  a 2 , 0  z  h}.(DS. 0) 46.  (Σ) ydydz + zdzdx + xdxdy, (Σ) l`a m˘a . tcu ’ a h`ınh ch´op gi´o . iha . n bo . ’ i c´ac m˘a . t ph˘a ’ ng x + y + z = a (a>0), x =0,y =0,z = 0. (D S. 0) 47.  (Σ) x 3 dydz + y 3 dzdx + z 3 dxdy, (Σ) l`a m˘a . tcˆa ` u x 2 + y 2 + z 2 = x. 174 Chu . o . ng 12. T´ıch phˆan h`am nhiˆe ` ubiˆe ´ n (DS. π 5 ) 48.  (Σ) x 3 dydz + y 3 dzdx + z 3 dxdy, (Σ) l`a m˘a . tcˆa ` u x 2 + y 2 + z 2 = a 2 . (D S. 12πa 5 5 ) 49.  (Σ) z 2 dxdy, (Σ) l`a m˘a . t elipxoid x 2 a 2 + y 2 b 2 + z 2 c 2 = 1. (DS. 0) Chı ’ dˆa ˜ n. Xem v´ıdu . 10, mu . cIII. 50.  (Σ) xdydz +ydzdx+zdxdy, (Σ) l`a m˘a . t elipxoid x 2 a 2 + y 2 b 2 + z 2 c 2 =1. (D s. 4πabc) 51.  (Σ) xdydz + ydzdx + zdxdy, (Σ) l`a biˆen h`ınh tru . x 2 + y 2  a 2 , −h  z  h.(D S. 6πa 2 h) 52.  (Σ) x 2 dydz + y 2 dzdx + z 2 dxdy, (Σ) l`a biˆen cu ’ a h`ınh lˆa . pphu . o . ng 0  x  a,0 y  a,0 z  a.(D S. 3a 4 ) D ˆe ’ ´ap du . ng cˆong th´u . c Stokes, ta lu . u´yla . iquyu . ´o . c Hu . ´o . ng du . o . ng cu ’ a chu tuyˆe ´ n ∂Σcu ’ am˘a . t (Σ) d u . o . . cquyu . ´o . cnhu . sau: Nˆe ´ umˆo . t ngu . `o . i quan tr˘a ´ cd ´u . ng trˆen ph´ıa d u . o . . ccho . ncu ’ am˘a . t(t´u . c l`a hu . ´o . ng t`u . chˆan d ˆe ´ ndˆa ` utr`ung v´o . ihu . ´o . ng cu ’ a vecto . ph´ap tuyˆe ´ n) th`ı khi ngu . `o . i quan s´at di chuyˆe ’ n trˆen ∂Σ theo hu . ´o . ng d ´o th`ı m˘a . t (Σ) luˆon luˆon n˘a ` m bˆen tr´ai. ´ Ap du . ng cˆong th´u . c Stokes d ˆe ’ t´ınh c´ac t´ıch phˆan sau 53.  C xydx + yzdy + xzdz, C l`a giao tuyˆe ´ ncu ’ am˘a . t ph˘a ’ ng 2x −3y + 4z − 12 = 0 v´o . i c´ac m˘a . t ph˘a ’ ng to . ad ˆo . .(DS. −7) [...]... y ’ o 1 1 n 1 2 n 3n−1 (−1)n 2n 0 (−1)n−1 3 3 (DS S = ) 2 (DS 2 ) 3 (DS Phˆn k`) a y n 1 ln2n 2 4 n 0 5 n 1 n(n + 5) 1 (DS 1 ) 1 − ln2 2 (DS 137 ) 30 0 ˜ o 13. 1 Chuˆ i sˆ du.o.ng o ´ 6 n 185 1 ,α (α + n)(α + n + 1) 1 7 n 3 8 n 1 n2 1 −4 (DS 2n + 1 + 1)2 n2 (n 0 (DS 25 ) 48 (DS 1) √ √ √ ( 3 n + 2 − 1 3 n + 1 + 3 n) 9 1 ) α+1 (DS 1 − √ 3 2) n 1 10 n 1 n(n + 3) (n + 6) 1 (DS 73 ) 1080 ’ ˜ ˜ ` ’... khi a > e) o a y nn 1 63 64 (DS Hˆi tu) o ˜ ’ a o ’ Trong c´c b`i to´n sau dˆy, h˜y khao s´t su hˆi tu cua chuˆ i d˜ a a a o a a a dˆu hiˆu du Cauchy cho nh` a o ´ e ’ n n (DS Hˆi tu) o 65 2n + 1 n 1 66 arc sin n 1 67 n 1 n 1 n+1 3n n 1 n (DS hˆi tu) o n2 (DS Hˆi tu) o ˜ ´ y e o Chu.o.ng 13 L´ thuyˆt chuˆ i 190 n5 68 n 1 69 n 1 70 n 3n + 2 4n + 3 3n n+5 n n+2 n +3 n n! √ n n 1 (DS Hˆi... k`) a y n 3n n2n 1 53 54 ˜ o 13. 1 Chuˆ i sˆ du.o.ng o ´ 55 n 189 1 · 3 · · · (2n − 1) 3n n! 1 (DS Hˆi tu) o π 2n (DS Hˆi tu) o (DS Hˆi tu) o n n(n + 1) 3n 1 (DS Hˆi tu) o n 73n (2n − 5)! 1 n (n + 1)! 2n n! 1 (DS Phˆn k`) a y n (2n − 1)!! n! 1 (DS Hˆi tu) o n n!(2n + 1)! (3n)! 1 n2 sin 56 n 1 57 58 59 60 61 nn sin 62 n 1 n! (DS Hˆi tu) o π 2n (DS Phˆn k`) a y n nn n!3n 1 n n!an... + 17 1 n 5 + 3( −1)n ’ ˜ (DS Hˆi tu) Chı dˆ n 2 o a 2n +3 1 n ln n n 1 32 33 34 (DS Hˆi tu) o 5 + 3( −1)n 8 ’ ˜ (DS Phˆn k`) Chı dˆ n ln n > 1 ∀ n > 2 a y a ln n (DS Hˆi tu) o n2 n 1 ’ ˜ ’ ’ o Chı dˆ n Su dung hˆ th´.c ln n < nα ∀ α > 0 v` n du l´.n a e u a ln n √ (DS Phˆn k`) a y 36 3 n n 1 35 n n5 √ 5 n 1 n 1 1 √ sin n n 1 37 38 39 n 40 (DS Hˆi tu) o (DS Hˆi tu) o n4 + 4n2 + 1 (DS... −2 2 √ ˜ ´ ´ hˆi tu nˆu a0 > 6 Do vˆy theo dˆu hiˆu so s´nh I chuˆ i o e a a e a o n2 e− n n 1 hˆi tu o ˜ ’ a o ’ o V´ du 3 Khao s´t su hˆi tu cua chuˆ i ı 1) n 1 2n + n2 , 3n + n 2) n 1 (n!)2 · (2n)! ’ Giai 1) Ta c´: o n+1 2 n an+1 + (n + 1) 3 +n 2 × n = n+1 = an 3 + (n + 1) 2 + n2 √ 2 n an = 3 n (n + 1)2 2n n+1 3+ n 3 2+ n 3n , × n2 1+ n 2 1+ n2 2n · n 1+ n 3 1+ an+1 2 2 √ ´ a a ’ a e T`... 13. 3 Chuˆ i l˜y th`.a 199 o u ’ ıa 13. 3.1 C´c dinh ngh˜ co ban 199 a - ` ’ a 13. 3.2 Diˆu kiˆn khai triˆn v` phu.o.ng ph´p khai e e e a ’n 201 triˆ e ˜ 13. 4 Chuˆ i Fourier 211 o ’ ıa 13. 4.1 C´c dinh ngh˜ co ban 211 a ˜ ´ ’ ` o ’ e o 13. 4.2 Dˆu hiˆu du vˆ su hˆi tu cua chuˆ i Fourier 212 a e ˜ ´ y e o Chu.o.ng 13. .. 1 ln n 1 n 1 n3n−1 1 n 1 √ 3 n+1 1 25 26 27 28 29 n 1 30 n 2n ˜ ’ a (DS Hˆi tu) Chı dˆ n nn > 2n ∀ n o 3 ˜ ` o ’ ˜ (DS Phˆn k`) Chı dˆ n So s´nh v´.i chuˆ i diˆu h`a o e a y a a o 1 +1 n (n + 2)2n 1 (DS Hˆi tu) o (DS Phˆn k`) a y (DS Hˆi tu) o (DS Hˆi tu) o ˜ o 13. 1 Chuˆ i sˆ du.o.ng o ´ 187 1 31 n 1 (n + 2)(n2 + 1) o (DS Hˆi tu) n 5n2 − 3n + 10 3n5 + 2n + 17 1 n 5 + 3( −1)n ’ ˜ (DS... biˆn 1 76 (DS 32 ) 5 Chu.o.ng 13 ˜ ´ L´ thuyˆt chuˆ i y e o ˜ o 13. 1 Chuˆ i sˆ du.o.ng 178 o ´ ’ ıa 13. 1.1 C´c dinh ngh˜ co ban 178 a ˜ ´ 13. 1.2 Chuˆ i sˆ du.o.ng 179 o o ˜ o ´ o a o o 13. 2 Chuˆ i hˆi tu tuyˆt d ˆi v` hˆi tu khˆng o e ´ o tuyˆt d ˆi 191 e ’ ıa 13. 2.1 C´c dinh ngh˜ co ban 191 a ˜ ´ ´ a a a e 13. 2.2 Chuˆ... du 6 Ch´.ng to r˘ng chuˆ i ı u o 2+ 10 26 n2 + 1 n3 − 1 5 7 − + − + ··· + + − 4 8 9 27 n2 n3 (*) ˜ hˆi tu, c`n chuˆ i o o o n2 + 1 n3 − 1 5 7 10 26 − + − + ··· + − + (**) 4 8 9 27 n2 n3 ˜ ˜ ´ ’ a a o a a o a thu du.o.c t` chuˆ i d˜ cho sau khi bo c´c dˆu ngo˘ c do.n l` chuˆ i phˆn a u k` y ’ ˜ ´ ’ Giai Sˆ hang tˆng qu´t cua chuˆ i (*) c´ dang o o a ’ o o 2+ an = n 2 + 1 n3 − 1 n+1 − = · 2 3. .. diˆu kiˆn khi 0 < p o e o e e ´ 3 (−1)n−1 sinp 19 n 1 n 5n + 1 √ , p > 0 n +3 n +3 (−1)n−1 √ ln n n+1 1 , p > 0 ˜ ’ a a ınh o ’ a Khao s´t d˘c t´ hˆi tu cua c´c chuˆ i (21 -32 ): o (−1)n+1 √ (DS Hˆi tu tuyˆt dˆi) o e o 21 ´ n3n n 1 (−1)n+1 n 1 1 (2n − 1 )3 2 ) 3 p 1 o o ` o e o e e (DS Hˆi tu tuyˆt dˆi khi p > ; hˆi tu c´ diˆu kiˆn khi 0 < p ´ 2 22 2 ) 3 n2 2 o o ` o e o e e (DS Hˆi tu . +1) 2 3 n+1 +(n +1) × 3 n + n 2 n + n 2 =  2+ (n +1) 2 2 n  3+ n +1 3 n × 1+ n 3 n 1+ n 2 2 n , n √ a n = 2 3 n       1+ n 2 2 n 1+ n 3 n · T`u . d ´o suy ra lim n→∞ a n+1 a n = 2 3 v`a. −c) 2 = R 2 .(DS. 8π 3 (a + b + c)R 3 ) 42.  (Σ) xdydz + ydzdx + zdxdy, (Σ) l`a m˘a . tcˆa ` u x 2 + y 2 + z 2 = R 2 . (D S. 4πR 3 ) 43.  (Σ) 4x 3 dydz +4y 3 dzdx − 6z 2 dxdy, (Σ) l`a biˆen. 191 13. 2.1 C´ac d i . nh ngh˜ıa co . ba ’ n 191 13. 2.2 Chuˆo ˜ id an dˆa ´ u v`a dˆa ´ uhiˆe . u Leibnitz . . . . 192 13. 3 Chuˆo ˜ il˜uy th`u . a 199 13. 3.1 C´ac d i . nh ngh˜ıa co . ba ’ n 199 13. 3.2