I 1 ii 1 i 1 2 i 1 i 1 20 2 i3 152 16 4 14400

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Unformatted text preview: 44 11 44 11 655 11 755 2 11 24 11 24 11 755 11 855 81 lim 4 n→ 18 lim 2 n→ n4 1 31 44 0.768 0.518 29. S 5 s5 1 11 855 11 955 2n3 n4 n n2 11 955 11 25 n2 81 1 4 1 7 1 8 1 8 1 9 1 9 1 10 0.746 0.646 11 655 31. lim n→ 81 n2 n 1 n4 4 18 n n 1 n2 2 1 n2 S 10 33. lim n→ n2 18 1 2 9 n 35. i 1 2i 1n 2i n2 i 1 12 10 1.02 1.002 1.0002 1.2 1 1 nn 1 2 n2 2 n n n 2 Sn S 100 S 1000 S 10,000 n 37. k 6k k 1 n3 1 6n 2 k n3 k 1 6 2n2 n2 k 3n 1 6 6 nn n3 3n 1 2n 6 3 1 nn 2 2 1 1 2n2 n2 Sn S 10 S 100 S 1000 S 10,000 n 1.98 1.9998 1.999998 1.99999998 16i n2 16 n i n2 i 1 16 n n 1 n2 2 n2 n2 n 1 n 39. lim n→ i 1 n→ lim n→ lim n→ lim 8 8 lim 1 n→ 8 184 Chapter 4 Integration n 41. lim n→ i 1 i n3 1 1 2 n→ lim 1n n3 i 1 i2 1 n→ lim 1n n3 n n→ 1 n 2n 6 lim 12 6 1 n n 1 3n 1 1 nn 1 n 2 1 n2 1 3 2 lim n→ 1 2n3 lim n→ 6 n 3n2 n3 n 43. lim n→ i 1 1 i n 2 n 2 lim n→ 1 n 1 i 1 1n i ni 1 2 lim n→ 1 n2 n 2n2 21 1 2 3 45. (a) 3 y (e) x sn Sn n 5 1.6 2.4 10 1.8 2.2 2 n 2 n 50 1.96 2.04 100 1.98 2.02 4n i n2i 1 2 1 x 1 3 (f) lim n→ i 1 i 1 n→ lim lim 1 n 4 n 2 (b) x 2 n 0 2 n n→ 4 nn 1 n2 2 2n n 4n i n2i 1 4 nn 1 n2 2 2n n 1 2 1 Endpoints: 0 <1 (c) Since y sn i n 1 n→ lim 2 2 <2 <...< n n n x is increasing, f mi n 1 2 2 <n n n 1 2 n→ n lim i 1 i 2 n 2 n n→ lim lim lim f xi on xi 1, xi . f xi f i 1 1 x 2 2 n n n→ 2i n 1, i i 1 2 1 n 2 n n→ (d) f Mi Sn i f xi on xi n xi n f xi 1 x i 1 f 2i 2 nn n i i 1 2 n 0 2 n 1 n 47. y 2x sn i 3 on 0, 1 . n Note: x n 1 n 3 2 y f 1 i n 2 n2i n 1 n i 1 2 i 1 i n 1n 1 n 1 n 2 3 Area 49. y x2 Sn i 1 n→ 3 2n 2n2 1 x lim s n 2 Note: x n i 1 1 2 3 2 on 0, 1 . n 1 n 2 y f 1 n3 i i n n 1 n i2 2 7 3 i n nn 2 1 2n 6n3 1 n 1 2 1 2 6 3 n 1 n2 2 3 1 1 x 1 2 3 Area n→ lim S n S ection 4.2 Area 185 51. y sn 16 n x 2 on 1, 3 . Note: x f1 i 1 2 n 1 2i n 2 y 18 2i n 2 n 4i2 n2 n 16 i 1 2 n 2n 15 ni 1 2 15n n 30 Area 4i n 1 2n 6 1 4 1 4 n n 70 3 4 n 1 9i n 27 n n n2 1 2n 27 2 1 n 1 2n 6 1 513 4 1 1 9nn 1 n 2 4nn 1 n 2 1 23 1 3 3 n 3i n 3 −1 14 12 10 8 6 4 2 x 1 2 3 4 nn n2 1 2n 30 8 n 6n2 n→ lim s n 8 3 53. y sn 64 n x3 on 1, 4 . Note: x f1 i 1 1 y 70 3i n 3 n 27i3 n3 n 64 i 1 3 n 60 50 40 30 20 10 −1 x 1 2 3 5 6 3n 63 ni 1 3 63n n 189 Area 27i2 n2 2 27 n2 n 1 n3 4 81 n 4n2 1 189 2 81 n 6n2 81 4 27 27 n 1 2 n 128.25 n→ lim s n 55. y x2 x3 on 1, 1 . Note: x 2 n 2 y Again, T n is neither an upper nor a lower sum. n Tn i n 1 f 1 i n 1 1 4i n 10i n 20 n2 10 1 4 2i n 2 n 4i2 n2 16i 2 n2 n 1 i 1 2i n 2 1 8i3 n3 20 n i n2 i 1 1 2 n 2i n 2 n 3 2 n 1 1 1 8i3 n3 32 n3 3 n 4 2 n 6i n 12i 2 n2 4n 1 ni 1 1 2n 6 41 x 1 2 i 1 32 n 2 i n3 i 1 16 n4 1 n2 n2 n 4 16 n 3 i n4 i 1 1 2 4 n n 4 Area nn 2 1 n 10 1 16 2 3 32 3 nn 1 n2 2 3 n→ lim T n 186 Chapter 4 Integration 57. f y Sn 3y, 0 ≤ y ≤ 2 n Note: y n 2 n 2 n 0 n 2 n 2i n 1 n 2 n 6 6 n 4 y f mi i 1 y i 1 f 12 n2 2i n 2 6 n 3 i 1 2 12 n i n2 i 1 Area n→ nn 1 6n x 2 4 6 lim S n n→ lim 6 6 2 59. f y Sn y 2, 0 ≤ y ≤ 3 n Note: y 3 n n i 1 3 n 2 0 3 n 6 4 2 y f i 1 3i n nn 3i n 1 27 2n 3 n 27 n 2 i n3 i 1 3n 2 9 3 n 1 2 n 10 8 6 27 n3 Area 61. g y Sn i n 1 n→ 1 2n 6 n→ 9 2n2 n2 9 2n2 1 9 27 2n 9 2n2 x 2 −2 −4 4 6 8 10 lim S n lim 9 4y2 n y3, 1 ≤ y ≤ 3. g1 41 2i n 2i n 4i n 10i n 6 2 n 2 Note: y y 1 4i2 n2 4i2 n2 10 i 1 2i n 1 8i3 n3 8 3 4 3 2 n 6i n 2 3n n 4 44 3 65. f x Let ci 12i2 n2 8i3 n3 4 nn n2 2 −4 −2 −2 −4 x 2n 41 ni 1 2n 3 ni 1 Area 63. f x Let ci x n n→ 10 n n 1 n 2 1 2n 6 1 8 n2 n 1 n2 4 2 lim S n x2 xi 3, 0 ≤ x ≤ 2, n xi 2 1 tan x, 0 ≤ x ≤ xi xi 2 , c1 x i 1 4 ,n 4 . 1 ,c 42 4 . , c2 4 1 ,c 21 f ci i 1 3 ,c 43 ci2 3 3 5 ,c 44 1 2 25 16 7...
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