11.4. T´ıch phˆan suy rˆo . ng 99 v`a +∞  a (αf(x)+βg(x))dx = α +∞  a f(x)dx + β +∞  a g(x)dx. 2) Cˆong th´u . c Newton-Leibnitz.Nˆe ´ u trˆen khoa ’ ng [a, +∞) h`am f(x) liˆen tu . cv`aF (x), x ∈ [a, +∞) l`a nguyˆen h`am n`ao d ´ocu ’ a n´o th`ı +∞  a f(x)dx = F(x)   +∞ a = F (+∞) −F(a) trong d ´o F (+∞) = lim x→+∞ F (x). 3) Cˆong th´u . cd ˆo ’ ibiˆe ´ n. Gia ’ su . ’ f(x), x ∈ [a,+∞) l`a h`am liˆen tu . c, ϕ(t), t ∈ [α, β] l`a kha ’ vi liˆen tu . cv`aa = ϕ(α)  ϕ(t) < lim t→β−0 ϕ(t)= +∞. Khi d ´o: +∞  a f(x)dx = β  α f(ϕ(t))ϕ  (t)dt. (11.27) 4) Cˆong th´u . c t´ıch phˆan t`u . ng phˆa ` n. Nˆe ´ u u(x)v`av(x), x ∈ [a, +∞) l`a nh˜u . ng h`am kha ’ vi liˆen tu . c v`a lim x→+∞ (uv)tˆo ` nta . i th`ı: +∞  a udv = uv   +∞ a − +∞  a vdu (11.28) trong d ´o uv   +∞ a = lim x→+∞ (uv) −u(a)v(a). 3. C´ac d iˆe ` ukiˆe . nhˆo . itu . 1) Tiˆeu chuˆa ’ n Cauchy.T´ıch phˆan +∞  a f(x)dx hˆo . itu . khi v`a chı ’ khi ∀ε>0, ∃b = b(ε)  a sao cho ∀b 1 >bv`a ∀b 2 >bta c´o:    b 2  b 1 f(x)dx    <ε. 100 Chu . o . ng 11. T´ıch phˆan x´ac d i . nh Riemann 2) Dˆa ´ uhiˆe . u so s´anh I. Gia ’ su . ’ g(x)  f(x)  0 ∀x  a v`a f(x), g(x) kha ’ t´ıch trˆen mo . id oa . n[a, b], b<+∞. Khi d´o : (i) Nˆe ´ u t´ıch phˆan +∞  a g(x)dx hˆo . itu . th`ı t´ıch phˆan +∞  a f(x)dx hˆo . itu . . (ii) Nˆe ´ u t´ıch phˆan +∞  a f(x)dx phˆan k`y th`ı t´ıch phˆan +∞  a g(x)dx phˆan k`y. 3) Dˆa ´ uhiˆe . u so s´anh II. Gia ’ su . ’ f(x)  0, g(x) > 0 ∀x  a v`a lim x→+∞ f(x) g(x) = λ. Khi d ´o: (i) Nˆe ´ u0<λ<+∞ th`ı c´ac t´ıch phˆan +∞  a f(x)dx v`a +∞  a g(x)dx d ˆo ` ng th`o . ihˆo . itu . ho˘a . cd ˆo ` ng th`o . i phˆan k`y. (ii) Nˆe ´ u λ = 0 v`a t´ıch phˆan +∞  a g(x)dx hˆo . itu . th`ı t´ıch phˆan +∞  a f(x)dx hˆo . itu . . (iii) Nˆe ´ u λ =+∞ v`a t´ıch phˆan +∞  a f(x)dx hˆo . itu . th`ı t´ıch phˆan +∞  a g(x)dx hˆo . itu . . D ˆe ’ so s´anh ta thu . `o . ng su . ’ du . ng t´ıch phˆan +∞  a dx x α   hˆo . itu . nˆe ´ u α>1, phˆan k`y nˆe ´ u α  1. (11.29) 11.4. T´ıch phˆan suy rˆo . ng 101 D - i . nh ngh˜ıa. T´ıch phˆan +∞  a f(x)dx du . o . . cgo . i l`a hˆo . itu . tuyˆe . td ˆo ´ inˆe ´ u t´ıch phˆan +∞  a |f(x)|dx hˆo . itu . v`a du . o . . cgo . i l`a hˆo . itu . c´o d iˆe ` ukiˆe . nnˆe ´ u t´ıch phˆan +∞  a f(x)dx hˆo . itu . nhu . ng t´ıch phˆan +∞  a |f(x)|dx phˆan k`y. Mo . i t´ıch phˆan hˆo . itu . tuyˆe . td ˆo ´ idˆe ` uhˆo . itu . . 3) T`u . dˆa ´ uhiˆe . u so s´anh II v`a (11.29) r´ut ra Dˆa ´ uhiˆe . u thu . . c h`anh. Nˆe ´ u khi x → +∞ h`am du . o . ng f(x) l`a vˆo c`ung b´e cˆa ´ p α>0 so v´o . i 1 x th`ı (i) t´ıch phˆan +∞  a f(x)dx hˆo . itu . khi α>1; (ii) t´ıch phˆan +∞  a f(x)dx phˆan k`y khi α  1. C ´ AC V ´ IDU . V´ı d u . 1. T´ınh t´ıch phˆan I = +∞  2 dx x 2 √ x 2 − 1 · Gia ’ i. Theo d i . nh ngh˜ıa ta c´o +∞  2 dx x 2 √ x 2 − 1 = lim b→+∞ b  2 dx x 2 √ x 2 − 1 · D ˘a . t x = 1 t , ta thu d u . o . . c 102 Chu . o . ng 11. T´ıch phˆan x´ac d i . nh Riemann I(b)= b  2 dx x 2 √ x 2 − 1 = 1/b  1/2 −dt t 2 · 1 t 2  1 t 2 − 1 = − 1/b  1/2 tdt √ 1 − t 2 = √ 1 − t 2    1/b 1/2 =  1 − 1 b 2 −  1 − 1 4 . T`u . d ´o suy r˘a ` ng I = lim b→+∞ I(b)= 2 − √ 3 2 .Nhu . vˆa . y t´ıch phˆan d ˜acho hˆo . itu . .  V´ı du . 2. Kha ’ o s´at su . . hˆo . itu . cu ’ a t´ıch phˆan +∞  1 2x 2 +1 x 3 +3x +4 dx. Gia ’ i. H`am du . ´o . idˆa ´ u t´ıch phˆan > 0 ∀x  1. Ta c´o f(x)= 2x 2 +1 x 3 +3x +4 = 2+ 1 x 2 x + 3 x + 4 x 2 · V´o . i x d u ’ l´o . n h`am f(x) c´o d´ang d iˆe . unhu . 2 x .Dod ´otalˆa ´ y h`am ϕ(x)= 1 x d ˆe ’ so s´anh v`a c´o lim x→+∞ f(x) ϕ(x) = lim x→+∞ (2x 2 +1)x x 2 +3x +4 =2=0. V`ı t´ıch phˆan ∞  1 dx x phˆan k`y nˆen theo dˆa ´ uhiˆe . u so s´anh II t´ıch phˆan d ˜a cho phˆan k`y.  V´ı du . 3. Kha ’ o s´at su . . hˆo . itu . cu ’ a t´ıch phˆan ∞  2 dx 3 √ x 3 − 12 · 11.4. T´ıch phˆan suy rˆo . ng 103 Gia ’ i. Ta c´o bˆa ´ td˘a ’ ng th ´u . c 1 3 √ x 3 − 1 > 1 x khi x>2. Nhu . ng t´ıch phˆan ∞  2 dx x phˆan k `y, do d ´o theo dˆa ´ uhiˆe . u so s´anh I t´ıch phˆan d ˜a cho phˆan k`y.  V´ı d u . 4. Kha ’ o s´at su . . hˆo . itu . v`a d ˘a . c t´ınh hˆo . itu . cu ’ a t´ıch phˆan +∞  1 sin x x dx. Gia ’ i. D ˆa ` u tiˆen ta t´ıch phˆan t`u . ng phˆa ` nmˆo . t c´ach h`ınh th´u . c +∞  1 sin x x dx = − cos x x    +∞ 1 − +∞  1 cos x x 2 dx = cos 1 − +∞  1 cos x x 2 dx. (11.30) T´ıch phˆan +∞  1 cos x x 2 dx hˆo . itu . tuyˆe . tdˆo ´ i, do d´on´ohˆo . itu . .Nhu . vˆa . y ca ’ hai sˆo ´ ha . ng o . ’ vˆe ´ pha ’ i (11.30) h˜u . uha . n. T`u . d ´o suy ra ph´ep t´ıch phˆan t `u . ng phˆa ` nd ˜a thu . . chiˆe . nl`aho . . pl´yv`avˆe ´ tr´ai cu ’ a (11.30) l`a t´ıch phˆan hˆo . itu . . Ta x´et su . . hˆo . itu . tuyˆe . td ˆo ´ i. Ta c´o |sin x|  sin 2 x = 1 − cos 2x 2 v`a do vˆa . y ∀b>1 ta c´o b  1 |sin x| x dx  1 2 b  1 dx x − 1 2 b  1 cos 2x x dx. (11.31) 104 Chu . o . ng 11. T´ıch phˆan x´ac d i . nh Riemann T´ıch phˆan th´u . nhˆa ´ to . ’ vˆe ´ pha ’ icu ’ a (11.31) phˆan k`y. T´ıch phˆan th ´u . hai o . ’ vˆe ´ pha ’ id ´ohˆo . itu . (diˆe ` ud´odu . o . . c suy ra b˘a ` ng c´ach t´ıch phˆan t`u . ng phˆa ` nnhu . (11.30)). Qua gi´o . iha . n (11.31) khi b → +∞ ta c´o vˆe ´ pha ’ i cu ’ a (11.31) dˆa ` nd ˆe ´ n ∞ v`a do d´o t´ıch phˆan vˆe ´ tr´ai cu ’ a (11.31) phˆan k`y, t ´u . c l`a t´ıch phˆan d ˜a cho hˆo . itu . c´o diˆe ` ukiˆe . n (khˆong tuyˆe . tdˆo ´ i).  B ` AI T ˆ A . P T´ınh c´ac t´ıch phˆan suy rˆo . ng cˆa . n vˆo ha . n 1. ∞  0 xe −x 2 dx (DS. 1 2 ) 2. ∞  0 dx x √ x 2 − 1 .(D S. π 6 ) 3. ∞  0 dx (x 2 +1) 2 .(DS. π − 2 8 ) 4. ∞  0 x sin xdx.(DS. Phˆan k`y) 5. ∞  −∞ 2xdx x 2 +1 .(D S. Phˆan k`y) 6. ∞  0 e −x sin xdx.(DS. 1 2 ) 7. +∞  2  1 x 2 −1 + 2 (x +1) 2  dx.(D S. 2 3 + 1 2 ln 3) 8. +∞  −∞ dx x 2 +4x +9 .(D S. π √ 5 5 ) 11.4. T´ıch phˆan suy rˆo . ng 105 9. +∞  √ 2 xdx (x 2 +1) 3 .(DS. 1 36 ). Chı ’ dˆa ˜ n. D ˘a . t x = √ t. 10. +∞  1 dx x √ x 2 + x +1 .(D S. ln  1+ 2 √ 3  ). Chı ’ dˆa ˜ n. D ˘a . t x = 1 t . 11. +∞  1 arctgx x 2 dx.(DS. π 4 + ln 2 2 ) 12. +∞  3 2x +5 x 2 +3x − 10 dx.(D S. Phˆan k`y) 13. ∞  0 e −ax sin bxdx, a>0. (DS. b a 2 + b 2 ) 14. +∞  0 e −ax cos bxdx, a>0. (DS. a a 2 + b 2 ) Kha ’ o s´at su . . hˆo . itu . cu ’ a c´ac t´ıch phˆan suy rˆo . ng cˆa . n vˆo ha . n 15. ∞  1 e −x x dx.(D S. Hˆo . itu . ) Chı ’ dˆa ˜ n. ´ Ap du . ng bˆa ´ td ˘a ’ ng th´u . c e −x x  e −x ∀x  1. 16. +∞  2 xdx √ x 4 +1 .(D S. Phˆan k`y) Chı ’ dˆa ˜ n. ´ Ap du . ng bˆa ´ td ˘a ’ ng th´u . c x √ x 4 +1 > x √ x 4 + x 4 ∀x  2. 17. +∞  1 sin 2 3x 3 √ x 4 +1 dx.(D S. Hˆo . itu . ) 106 Chu . o . ng 11. T´ıch phˆan x´ac d i . nh Riemann 18. +∞  1 dx √ 4x +lnx .(D S. Phˆan k `y) 19. +∞  1 ln  1+ 1 x  x α dx.(DS. Hˆo . itu . nˆe ´ u α>0) 20. +∞  0 xdx 3 √ x 5 +2 .(D S. Hˆo . itu . ) 21. +∞  1 cos 5x − cos 7x x 2 dx.(DS. Hˆo . itu . ) 22. +∞  0 xdx 3 √ 1+x 7 .(DS. Hˆo . itu . ) 23. +∞  0 √ x +1 1+2 √ x + x 2 dx.(DS. Hˆo . itu . ) 24. ∞  1 1 √ x (e 1/x −1)dx.(DS. Hˆo . itu . ) 25. ∞  1 x + √ x +1 x 2 +2 5 √ x 4 +1 dx.(D S. Phˆan k`y) 26. ∞  3 dx  x(x − 1)(x −2) .(D S. Hˆo . itu . ) 27 ∗ . ∞  0 (3x 4 − x 2 )e −x 2 dx.(DS. Hˆo . itu . ) Chı ’ dˆa ˜ n. So s´anh v´o . i t´ıch phˆan hˆo . itu . +∞  0 e − x 2 2 dx (ta . i sao ?) v`a ´ap du . ng dˆa ´ uhiˆe . u so s´anh II. 11.4. T´ıch phˆan suy rˆo . ng 107 28 ∗ . +∞  5 ln(x −2) x 5 + x 2 +1 dx.(D S. Hˆo . itu . ) Chı ’ dˆa ˜ n. ´ Ap du . nng hˆe . th ´u . c lim t→+∞ ln t t α =0∀α>0 ⇒ lim x→+∞ ln(x −2) x α =0∀α>0. T`u . d ´o so s´anh t´ıch phˆan d˜a c h o v ´o . i t´ıch phˆan hˆo . itu . +∞  5 dx x α , α>1. Tiˆe ´ pd ˆe ´ n´apdu . ng dˆa ´ uhiˆe . u so s´anh II. 11.4.2 T´ıch phˆan suy rˆo . ng cu ’ a h`am khˆong bi . ch˘a . n 1. Gia ’ su . ’ h`am f(x) x´ac d i . nh trˆen khoa ’ ng [a, b) v`a kha ’ t´ıch trˆen mo . i d oa . n[a, ξ], ξ<b.Nˆe ´ utˆo ` nta . i gi´o . iha . nh˜u . uha . n lim ξ→b−0 ξ  0 f(x)dx (11.32) th`ı gi´o . iha . nd ´odu . o . . cgo . i l`a t´ıch phˆan suy rˆo . ng cu ’ a h`am f(x) trˆen [a, b) v`a k´yhiˆe . u l`a: b  a f(x)dx. (11.33) Trong tru . `o . ng ho . . p n`ay t´ıch phˆan suy rˆo . ng (11.33) d u . o . . cgo . il`at´ıch phˆan hˆo . itu . .Nˆe ´ u gi´o . iha . n (11.32) khˆong tˆo ` nta . ith`ıt´ıch phˆan suy rˆo . ng (11.33) phˆan k `y. D i . nh ngh˜ıa t´ıch phˆan suy rˆo . ng cu ’ a h`am f(x)x´acdi . nh trˆen khoa ’ ng (a, b]d u . o . . c ph´at biˆe ’ utu . o . ng tu . . . Nˆe ´ u h`am f(x) kha ’ t´ıch theo ngh˜ıa suy rˆo . ng trˆen c´ac khoa ’ ng [a, c) v`a (c, b] th`ı h`am d u . o . . cgo . i l`a h`am kha ’ t´ıch theo ngh˜ıa suy rˆo . ng trˆen 108 Chu . o . ng 11. T´ıch phˆan x´ac d i . nh Riemann doa . n[a, b] v`a trong tru . `o . ng ho . . p n`ay t´ıch phˆan suy rˆo . ng d u . o . . c x´ac d i . nh bo . ’ id ˘a ’ ng th ´u . c: b  a f(x)dx = c  a f(x)dx + b  c f(x)dx. 2. C´ac cˆong th´u . cco . ba ’ n 1) Nˆe ´ u c´ac t´ıch phˆan b  a f(x)dx v`a b  a g(x)dx hˆo . itu . th`ı ∀α, β ∈ R ta c´o t´ıch phˆan b  a [αf(x)+βg(x)]dx hˆo . itu . v`a b  a [αf(x)+βg(x)]dx = α b  a f(x)dx + β b  a g(x)dx. 2) Cˆong th´u . c Newton-Leibnitz. Nˆe ´ u h`am f(x), x ∈ [a, b)liˆen tu . c v`a F (x) l`a mˆo . t nguyˆen h`am n`ao d ´ocu ’ a f trˆen [a, b) th`ı: b  a f(x)dx = F (x)   b−0 a = F(b − 0) − F (a), F (b −0) = lim x→b−0 F (x). 3) Cˆong th´u . cd ˆo ’ ibiˆe ´ n. Gia ’ su . ’ f(x) liˆen tu . c trˆen [a, b) c`on ϕ(t), t ∈ [α, β) kha ’ vi liˆen tu . cv`aa = ϕ(α)  ϕ(t) < lim t→β−0 ϕ(t)=b. Khi d ´o : b  a f(x)dx = β  α f[ϕ(t)]ϕ  (t)dt. [...]... y = − x + , y = − x + 5 biˆn th`nh c´c du.`.ng th˘ng v = , v = 5 3 3 3 3 ∗ th`nh miˆn D∗ = [ 3, 1] × 7 , 5 Dˆ d`ng thˆy ˜ a ` ´ ’ e a Do d´ miˆn D tro a e o ` e 3 D(x, y) 3 ` r˘ng a = − Do d´ theo cˆng th´.c (12.5): o o u D(u, v) 4 3 1 u+ v − 4 4 (y − x)dxdy = 3 3 − u+ v 4 4 3 dudv 4 D∗ D 5 3 ududv = 4 = D∗ 4 3 udu = −8 4 dv 7 /3 3 ’ ´ ` Nhˆn x´t Ph´p dˆi biˆn trong t´ phˆn hai l´.p nh˘m muc d´ch... 12.1 .3 Mˆt v`i u.ng dung trong h` hoc 121 o a ´ 12.2 T´ phˆn 3- l´.p 133 ıch a o ` ınh o o e 12.2.1 Tru.`.ng ho.p miˆn h` hˆp 133 ` o e 12.2.2 Tru.`.ng ho.p miˆn cong 1 34 12.2 .3 136 12.2 .4 Nhˆn x´t chung 136 a e o 12 .3 T´ phˆn d u.`.ng 144 ıch a ’ ıa 12 .3. 1 C´c dinh ngh˜ co ban 144 a ... biˆn 130 xdxdy; y = x3, x + y = 2, x = 0 30 7 ) 15 (DS D 31 5 (DS 20 ) 6 xdxdy; xy = 6, x + y − 7 = 0 D 3 (DS 1 ) 5 y 2xdxdy; x2 + y 2 = 4, x + y − 2 = 0 32 D 33 (x + y)dxdy; 0 y π, 0 x sin y (DS 5π ) 4 D 34 sin(x + y)dxdy; x = y, x + y = π , y = 0 2 (DS 1 ) 2 D 2 e−y dxdy; D l` tam gi´c v´.i dınh O(0, 0), B(0, 1), A(1, 1) a a o ’ 35 D (DS − 1 1 + ) 2e 2 xydxdy; D l` h`nh elip 4x2 + y 2 a ı 36 4 (DS... 0, y = 37 √ 2ax − x2 (DS 4a5 ) 5 D xdxdy ; y = x, x = 2, x = 2y x2 + y 2 38 (DS 1 π − 2arctg ) 2 2 D √ x + ydxdy; x = 0, y = 0, x + y = 1 39 (DS 2 ) 5 (DS 4 4 ) 15 D (x − y)dxdy; y = 2 − x2, y = 2x − 1 40 D 41 (x + 2y)dxdy; y = x, y = 2x, x = 2, x = 3 D 1 (DS 25 ) 3 12.1 T´ phˆn 2-l´.p ıch a o 42 131 xdxdy; x = 2 + sin y, x = 0, y = 0, y = 2π (DS 9π ) 2 D xydxdy; (x − 2)2 + y 2 = 1 43 (DS 4 ) 3 D dxdy... xdx (DS −0, 25) 0 3 √ 4 6 xdx x2 − 4 (DS 2√ 4 125) 3 2 2 dx (x − 1)2 7 (DS Phˆn k`) a y 0 2 xdx x2 − 1 8 (DS Phˆn k`) a y −2 2 9 x3 dx √ 4 − x2 (DS 16 ’ ˜ ) Chı dˆ n D˘t x = 2 sin t a a 3 0 0 10 −1 e1/x dx x3 2 (DS − ) e 11 .4 T´ phˆn suy rˆng ıch a o 1 11 e1/x dx x3 115 (DS Phˆn k`) a y 0 1 dx 12 x(1 − x) 0 (DS π) b dx ; a < b (x − a)(b − x) 13 a (DS π) 1 x ln2 xdx 14 (DS 1 ) 4 0 ’ a o ’ a ıch... 128 4 9 4 4 dy 0 f(−)dx (DS dx y 0 f(−)dy (DS f (−)dx) √ − 2−x2 0 1 y f(−)dy (DS dy x 0 y 2 1 dy 1 dy 0 dx 12 y−1 1 x+1 1 11 2 1−x2 dx −1 f (−)dy) 2 √ 10 x fdx (DS 1/y 1−y 2 √ 2 f dx + 0 dx 1/2 f dy + 13 2x dx 0 (x − y + 1)dy 14 y y3 dx x2 + y 2 (DS 6π) (x + 2y)dx (DS −11, 2) dy −2 0 y2 0 15 dy 2 0 5 16 5−x dx 0 4 + x + ydy (DS 506 ) 15 0 4 17 2 dy (x + y)2 dx 3 (DS 25 ) 24 1 √ 2 ax a dx 0 1 ) 3 x 4. .. x2) √ √ ∼ = √ = ϕ(x) 3 x sin x (x→0+0) x x 1 dx ´ √ hˆi tu nˆn theo dˆu hiˆu so s´nh II, t´ch phˆn o e a e a ı a 3 x V` t´ phˆn ı ıch a 0 d˜ cho hˆi tu a o ıch a a Chu.o.ng 11 T´ phˆn x´c dinh Riemann 1 14 ` ˆ BAI TAP T´ c´c t´ phˆn suy rˆng sau ınh a ıch a o 6 1 √ (DS 6 3 2) dx (4 − x)2 3 2 2 dx 2 (x − 1)2 3 0 (DS 6) e dx x ln x 3 (DS Phˆn k`) a y 1 2 4 x2 dx − 4x + 3 (DS Phˆn k`) a y 0... α > 0 ⇒ c´ thˆ lˆy a x→0+0 1 |lnx| 1 ’ α = ch˘ng han ⇒ √ < 3/ 4 a 4 x x 1 sin x dx x2 24 (DS Phˆn k`) a y 0 2 √ 25 dx x − x3 (DS Hˆi tu) o 0 2 26 x2 (x − 2) dx − 3x2 + 4 (DS Phˆn k`) a y 1 1 dx 27 x(ex − e−x ) 0 2 16 + x4 dx 16 − x4 28 o (DS Hˆi tu) (DS Hˆi tu) o 0 1√ ex − 1 dx sin x 29 (DS Hˆi tu) o 0 1 3 ln(1 + x) dx 1 − cos x 30 0 (DS Phˆn k`) a y Chu.o.ng 12 ´ ` T´ phˆn h`m nhiˆu biˆn... x 1 + x2dx = 13 3 2 dy = 0 x/2 ` ˆ BAI TAP ’ ıch a T` cˆn cua t´ phˆn hai l´.p ım a o ` f (x, y)dxdy theo miˆn D gi´.i e o D ’ a ’ a han bo.i c´c du.`.ng d˜ chı ra (Dˆ ng˘n gon ta k´ hiˆu f (x, y) = f (−)) o a ’ e ´ y e 1 x = 3, x = 5, 3x − 2y + 4 = 0, 3x − 2y + 1 = 0 3x +4 5 5 (DS dx 3 f (−)dy) 3x+1 5 2 x = 0, y = 0, x + y = 2 2 (DS 2−x dx 0 f (−)dy) 0 12.1 T´ phˆn 2-l´.p ıch a o 3 x2 + y 2 1,... a e u o a o 3 44 45 ydxdy; x = R(t − sin t), y = R(1 − cos t), 0 t 2π (l` miˆn a ` e D 5 ’ o ’ gi´.i han bo.i v`m cua xicloid.) (DS πR3 ) o 2 y=f (x) 2πR ˜ ’ a Chı dˆ n ydxdy = dx 0 D ydy 0 ’ ı a Chuyˆn sang toa dˆ cu.c v` t´nh t´ch phˆn trong toa dˆ m´.i e o o o a ı πR4 46 ) (x2 + y 2 )dxdy; D : x2 + y 2 R2 , y 0 (DS 4 D π (e − 1)) 4 ex 2 +y 2 dxdy; D : x2 + y 2 1, x 0, y ex 47 2 +y 2 dxdy; . 1 34 12.2 .3 136 12.2 .4 Nhˆa . nx´etchung 136 12 .3 T´ıch phˆan d u . `o . ng 144 12 .3. 1 C´ac d i . nh ngh˜ıa co . ba ’ n 144 12 .3. 2 T´ınh t´ıch phˆan d u . `o . ng 146 12 .4 T´ıch phˆan m˘a . t 158 12 .4. 1. sau. 1. 6  2 dx 3  (4 − x) 2 .(DS. 6 3 √ 2) 2. 2  0 dx 3  (x −1) 2 .(DS. 6) 3. e  1 dx x ln x .(D S. Phˆan k`y) 4. 2  0 dx x 2 − 4x +3 .(D S. Phˆan k`y) 5. 1  0 x ln xdx.(DS. −0, 25) 6. 3  2 xdx 4 √ x 2 −. 118 12.1 .3 Mˆo . tv`ai´u . ng du . ng trong h`ınh ho . c 121 12.2 T´ıch phˆan 3- l´o . p 133 12.2.1 Tru . `o . ng ho . . pmiˆe ` n h`ınh hˆo . p 133 12.2.2 Tru . `o . ng ho . . pmiˆe ` ncong 1 34 12.2.3