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 œœ œ "" ## 1 x dx 1 square unit ʱœ ' ±" " ab kk 21. The area of the triangular peak is A bh (2)(1) 1. œ The area of the rectangular base is S w (2)(1) 2. œj œ œ Then the total area is 3 2 3 square units ' ±" " 22. y 1 1 x y 1 1 x œ² ± Ê ±œ ± ÈÈ (y 1) 1 x x (y 1) 1, a circle with ±Ê² ±œ # # center ( ) and radius of 1 y 1 1 x is the !ß" Ê œ ² ± È # upper semicircle. The area of this semicircle is A r (1) . The area of the rectangular base œ # 11 1 is A w (2)(1) 2. Then the total area is 2 œ ² 1 # 1 1 x dx 2 square units ʲ ± œ ² ' ±" " # # Š‹ È 1 23. dx (b)( ) 24. 4x dx b(4b) 2b '' ! ! " " # # # b b xb b 22 4 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 ± ± "" ## ### ab 27. x dx 28. x dx 3 " !Þ& # #Þ& # " È Š‹ È œ ± œ 2 (1) (2.5) (0.5) # # 29. d 30. r dr 24 1 1 1 11 # &# # # # )) œ± œ œ ± œ (2 ) 3 52 2 # È È Š‹ Š‹ ÈÈ 31. x dx 32. s ds 0.009 " ! ( !Þ$ # # È È $ $ $ $ œœ œ œ 7 33 3 7 (0.3) 33. t dt 34. d ! ! "Î# Î# # # " # ˆ‰ " # # $ $ $ 32 4 3 4 1 1 1 35. x dx 36. x dx a a a a a (2a) a3 a a 3a # $ # # # œ œ ± œ # # # È È 37. 38. 9b ! ! # #$ $ È È $ $ $ $ b b b 3 b (3b) œ œ 39. 7 dx 7(1 3) 14 40. 2 dx 2 ( ) 2 2 $ ! " ± œ ± œ ± # ± ! œ ± 2 È 41. 5x dx 5 x dx 5 10 42. dx x dx 1 ' ' !! $ $ && 22 20 x 53 1 6 88 8 1 6 œ œ ±œ œ œ œ ’“ 43. (2t 3) dt 2 t dt 3 dt 2 3(2 0) 4 6 2 ' !" ! " ± œ ± œ ± ± ± œ±œ± 44. t 2 dt t dt 2 dt 2 2 0 1 2 1 ' ! È È 2 2 0 È È –— ± œ ± œ ± ± ± œ±œ± # # 45. 1 dz 1 dz dz 1 dz z dz 1[1 2] ' ' ' # # " " " # # # # # # # " ²œ ² œ ± œ ± ±± œ ± " ± zz 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 ' ' ' $$ $ ! $ ! $ ! ± œ ± ± œ ± ± ± ± œ ± ² œ 30 47. 3u du 3 u du 3 3 3 3 7 ''' ' ""! ! " # # " œ œ ± œ ±±± œ œ ”• 0 21 7 3
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Section 5.3 The Definite Integral 307 48. 24u du 24 u du 24 24 24 7 '' ' ' "Î# "Î# ! ! " " " "Î# ### # œœ± œ ± œ œ –— ”• ’“ 1 33 3 $ " # $ ˆ‰ 7 8 49. 3x x 5 dx 3 x dx x dx 5 dx 3 5[2 0] (8 2) 10 0 ' ' !! ! ! ## # # ab ² ± œ ² ± œ ±²±±± œ ² ± œ 20 $$ 50. 3x x 3x x 3 x dx 5 dx ' ' ' "! ! ! ! !" " " " ²± œ± ² ± 3 5(1 0) 5 œ ± ±²±±±œ ± ± œ Š‹ 10 3 7 # # 51. Let x and let x 0, x x, ?? œœ œ œ b0 b nn ± x 2 x x (n 1) x, x n x b. # œß á ßœ ± œ œ ? " Let the c 's be the right end-points of the subintervals k c x , c x , and so on. The rectangles Êœ œ "" defined have areas: f(c ) x f( x) x 3( x) x 3( x) " #$ œ f ( c )xf ( 2x )x3 ( ( 2 ) (x ) # $ ? ? ? ? œ f ( c ( 3x ( ( 3 ) ) $ $ ? ? ? ? œ ã f ( c ( nx ( ( n ) ) n ? ? ? ? œ $ Then S f(c ) x 3k ( x) nk k1 oeoe 3( x) k 3 ? $# ²² ! Š n oe b n6 n ( n1 ) ( 2 ) $ $ 2 3x dx lim 2 b . œ ² ²Ê œ ² ²œ b3 b $ $ # # ! ' n Ä_ 52. Let x and let x 0, x x, œ œ b ± x 2 x x (n 1) x, x n x b. # á ± œ œ ? " Let the c 's be the right end-points of the subintervals k c x , c x , and so on. The rectangles œ defined have areas: f ( c ) x f ) x ) " ? 1 ? ? 1 ? œ f(c ) x f(2 x) x (2 x) x (2) ( x) # $ ? 1 ? ? 1 ? œ f(c ) x f(3 x) x (3 x) x (3) ( x) $ $ ? 1 ? ? 1 ? œ ã f(c ) x f(n x) x (n x) x (n) ( x) n ? 1 ? ? 1 ? œ $ Then S f(c ) x k ( x) ?1 ? ) k 1? 1 ! Š n oe b n ( ) ( 2 ) $ $ 2 2 . œ ² œ ² 11 1 b 6n n n 3 b $ # # ! # ' 1 n
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308 Chapter 5 Integration 53. Let x and let x 0, x x, ?? œœ œ œ b0 b nn ± !" x 2 x x (n 1) x, x n x b.
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This note was uploaded on 09/20/2010 for the course MATHEMATIC 09991051 taught by Professor Dr.maenshadeed during the Fall '10 term at Norwegian Univ. of Science & Technology.

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

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