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ISM_T11_C05_B - Section 5.3 The Definite Integral 20 The...

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Section 5.3 The Definite Integral 305 20. The area of the triangle is A bh (2)(1) 1 œ œ œ " " # # 1 x dx 1 square unit Ê œ ' " " a b k k 21. The area of the triangular peak is A bh (2)(1) 1. œ œ œ " " # # The area of the rectangular base is S w (2)(1) 2. œ j œ œ Then the total area is 3 2 x dx 3 square units Ê œ ' " " a b k k 22. y 1 1 x y 1 1 x œ Ê œ È È # # (y 1) 1 x x (y 1) 1, a circle with Ê œ Ê œ # # # # center ( ) and radius of 1 y 1 1 x is the !ß " Ê œ È # upper semicircle. The area of this semicircle is A r (1) . The area of the rectangular base œ œ œ " " # # # # # 1 1 1 is A w (2)(1) 2. Then the total area is 2 œ j œ œ 1 # 1 1 x dx 2 square units Ê œ ' " " # # Š È 1 23. dx (b)( ) 24. 4x dx b(4b) 2b ' ' ! ! " " # # # b b x b b 2 2 4 œ œ œ œ
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306 Chapter 5 Integration 25. 2s ds b(2b) a(2a) b a 26. 3t dt b(3b) a(3a) b a ' ' a a b b 3 œ œ œ œ " " " " # # # # # # # # # a b 27. x dx 28. x dx 3 ' ' " !Þ& # #Þ& # # # # # " È Š È œ œ œ œ 2 (1) (2.5) (0.5) 29. d 30. r dr 24 ' ' 1 1 1 1 1 # & # # # # # # # ) ) œ œ œ œ (2 ) 3 5 2 2 È È Š Š È È 31. x dx 32. s ds 0.009 ' ' " ! ( !Þ$ # # È Š È œ œ œ œ 7 3 3 3 7 (0.3) 33. t dt 34. d ' ' ! ! "Î# Î# # # " # œ œ œ œ ˆ ‰ ˆ 3 24 3 4 1 1 ) ) 35. x dx 36. x dx a ' ' a a a a (2a) a 3a a 3a # $ # # # # # # œ œ œ œ È Š È 37. x dx 38. x dx 9b ' ' ! ! # # $ $ È Š È b b b 3 3 3 b (3b) œ œ œ œ 39. 7 dx 7(1 3) 14 40. 2 dx 2 ( ) 2 2 ' ' $ ! " œ œ œ # ! œ 2 È È È 41. 5x dx 5 x dx 5 10 42. dx x dx 1 ' ' ' ' ! ! $ $ # # # # & & " " 2 2 2 0 x 5 3 16 8 8 8 16 œ œ œ œ œ œ œ 43. (2t 3) dt 2 t dt 3 dt 2 3(2 0) 4 6 2 ' ' ' ! " ! " # # 2 2 2 0 œ œ œ œ 44. t 2 dt t dt 2 dt 2 2 0 1 2 1 ' ' ' ! ! ! # # È È È Š È 2 2 2 2 0 Š È È È È œ œ œ œ 45. 1 dz 1 dz dz 1 dz z dz 1[1 2] ' ' ' ' ' # # # # " " " " " # # # # # # # # # " " " ˆ ˆ ‰ œ œ œ œ " œ z z 2 1 3 7 4 46. (2z 3) dz 2z dz 3 dz 2 z dz 3 dz 2 3[0 3] 9 9 0 ' ' ' ' ' $ $ $ ! $ ! ! ! $ ! # # œ œ œ œ œ 3 0 47. 3u du 3 u du 3 u du u du 3 3 3 7 ' ' ' ' " " ! ! # # # " # # # # " œ œ œ œ œ œ Š ˆ ‰ 2 0 0 2 1 7 3 3 3 3 3 3 3
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Section 5.3 The Definite Integral 307 48. 24u du 24 u du 24 u du u du 24 24 7 ' ' ' ' "Î# "Î# ! ! " " " "Î# # # # # œ œ œ œ œ 1 3 3 3 ˆ ‰ ˆ 7 8 49. 3x x 5 dx 3 x dx x dx 5 dx 3 5[2 0] (8 2) 10 0 ' ' ' ' ! ! ! ! # # # # # # # # a b œ œ œ œ 2 0 2 0 3 3 50. 3x x 5 dx 3x x 5 dx 3 x dx x dx 5 dx ' ' ' ' ' " ! ! ! ! ! " " " " # # # a b a b œ œ 3 5(1 0) 5 œ œ œ Š Š ˆ 1 0 1 0 3 7 3 3 # # # # 51. Let x and let x 0, x x, ? ? œ œ œ œ b 0 b n n ! " x 2 x x (n 1) x, x n x b. # œ ß á ß œ œ œ ? ? ? n n Let the c 's be the right end-points of the subintervals k c x , c x , and so on. The rectangles Ê œ œ " " # # defined have areas: f(c ) x f( x) x 3( x) x 3( x) " # $ ? ? ? ? ? ? œ œ œ f(c ) x f(2 x) x 3(2 x) x 3(2) ( x) # # # $ ? ? ? ? ? ? œ œ œ f(c ) x f(3 x) x 3(3 x) x 3(3) ( x) $ # # $ ? ? ? ? ? ? œ œ œ ã f(c ) x f(n x) x 3(n x) x 3(n) ( x) n ? ? ? ? ? ? œ œ œ # # $ Then S f(c ) x 3k ( x) n k n n k 1 k 1 œ œ ! ! ? ? # $ 3( x) k 3 œ œ ? $ # ! Š ‹ Š n k 1 b n 6 n(n 1)(2n 1) 2 3x dx lim 2 b . œ Ê œ œ b 3 b 3 n n n n b # # " " ! # $ ˆ ˆ ' n Ä _ 52. Let x and let x 0, x x, ? ? œ œ œ œ b 0 b n n ! " x 2 x x (n 1) x, x n x b. # œ ß á ß œ œ œ ? ? ? n n Let the c 's be the right end-points of the subintervals k c x , c x , and so on. The rectangles Ê œ œ " " # # defined have areas: f(c ) x f( x) x ( x) x ( x) " # $ ? ? ? 1 ? ? 1 ? œ œ œ f(c ) x f(2 x) x (2 x) x (2) ( x) # # # $ ? ? ? 1 ? ? 1 ? œ œ œ f(c ) x f(3 x) x (3 x) x (3) ( x) $ # # $ ? ? ? 1 ? ? 1 ? œ œ œ ã f(c ) x f(n x) x (n x) x (n) ( x) n ? ? ? 1 ? ? 1 ? œ œ œ # # $ Then S f(c ) x k ( x) n k n n k 1 k 1 œ œ ! !
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