ISM_T11_C05_A - CHAPTER 5 INTEGRATION 5.1 ESTIMATING WITH...

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CHAPTER 5 INTEGRATION 5.1 ESTIMATING WITH FINITE SUMS 1. f x x ab œ # Since f is increasing on , we use left endpoints to obtain Ò!ß "Ó lower sums and right endpoints to obtain upper sums. (a) x and x i x a lower sum is ˜œ œ œ˜œ Ê † œ !± œ "±! " " " " " ## # # # # # ) œ! " # # # i ii i ! ˆ‰ Š‹ (b) x and x i x a lower sum is œ ± ± œ † œ " " " " " $ " ( ( %% % # % ) $ # œ! $ # # ### i i 44 4 4 4 ! (c) x and x i x an upper sum is +1 œ † œ œ " " " " & # # # # # ) œ # # # i i1 2 ! (d) x and x i x an upper sum is +1 œ ± ± œ † œ " " " " " $ " $! "& % # % " '$ # œ" % # # i i 4 4 4 ! ˆ ‰ 2. f x x œ $ Since f is increasing on , we use left endpoints to obtain Ò!ß "Ó lower sums and right endpoints to obtain upper sums. (a) x and x i x a lower sum is œ œ " " " " " # # # # # " ' œ! " $ $ $ i i ! (b) x and x i x a lower sum is œ ± ± œ œ " " " " " $ $' * % % % # #&' '% œ! $ $ $ $$$ i i 4 4 4 ! (c) x and x i x an upper sum is +1 œ " " " " " * * # # # # # # )" ' œ $ $ $ i 2 ! (d) x and x i x an upper sum is +1 œ ± ± œœ œ " " " " " $ "!! #& % % % # #&' '% œ" % $ $ i i 4 4 4 !
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294 Chapter 5 Integration 3. f x ab œ " x Since f is decreasing on , we use left endpoints to obtain Ò!ß "Ó upper sums and right endpoints to obtain lower sums. (a) x and x i x i a lower sum is ˜ œ œ# œ"± ˜ œ"±# Ê †#œ# ± œ &±" " " " "' # $& " & œ" # i i x ! ˆ‰ i (b) x and x i x i a lower sum is œ% œ"± ˜ œ"± Ê †"œ" ± ± ± œ & ± " " """" ( ( % #$%& ' ! œ" % i i x ! i (c) x and x i x i an upper sum is †#œ# "± œ "" ) # $$ œ! " i i x ! i (d) x and x i x i an upper sum is †"œ" "± ± ± œ " " " " #& % #$% " # œ! $ i i x ! i 4. f x x œ%² # Since f is increasing on and decreasing on , we use Ò²#ß !Ó Ò!ß #Ó left endpoints on and right endpoints on to obtain Ò²#ß !Ó Ò!ß #Ó lower sums and use right endpoints on and left endpoints Ò²#ß !Ó on to obtain upper sums. Ò!ß #Ó (a) x and x i x i a lower sum is œ²#± ˜ œ²#±# Ê #† %² ²# ±#† %²# œ! #± ±# # # # i a b (b) x and x i x i a lower sum is x x ˜œ œ" œ² #±˜œ² #±Ê " ## % œ! œ$ "% ii i !! ˆ‰ˆ‰ œ" %² ²# ± %² ²" ± %²" ± %²# œ' ab a ba b (c) x and x i x i a upper sum is #† %² ! ±#† %²! œ"' # # # i a b (d) x and x i x i a upper sum is x x " % œ" œ# #$ i œ" %²² " ±%²! ±%²! ±%²" œ" % a b # ### 5. f x x œ # Using 2 rectangles x f f ʘ œ œ Ê ± "±! " " " $ # % % ˆ‰ ˆ‰ œ± œ œ $ " ! & #% % $ # " ' Š‹ Using 4 rectangles x œ " %% Ê ±±± " "$&( % )))) ˆ‰ ˆ‰ ˆ‰ ˆ‰ ffff œ œ " # " % ' % # #
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Section 5.1 Estimating with Finite Sums 295 6. f x x ab œ $ Using 2 rectangles x f f ʘ œ œ Ê ± "±! " " " $ ## # % % ˆ‰ ˆ‰ ˆ‰ œ± œ œ "" $ # ) ( #% % # ' % $ # $$ Š‹ Using 4 rectangles x œ " %% Ê ±±± " "$&( % )))) ˆ‰ ˆ‰ ˆ‰ ˆ‰ ffff œœ œ œ " " ²$ ²& ²( %*' "#% $" % ) %†) ) "#) $$$$ $ 7. f x œ " x Using 2 rectangles x f f œ#Ê# # ± % &±" # ab ab œ# ± œ "" $ #% # Using 4 rectangles x œ" % Ê " $&(* #### œ " ±±± œ œ œ "%)) %*' %*' $&(* $ & ( *& ( *$ " & 8. f x x œ%² # Using 2 rectangles x f f œ#Ê# ²" ± " #± ±# # ab a b œ#$±$ œ"# Using 4 rectangles x % Ê "²±²± ± ˆ‰ ˆ‰ ˆ ‰ ˆ fff f $ ""$ # ### œ" %² ² ± %² ² ± %² Š Š ˆ ˆ $ # # œ "' ² † # ± † # œ "' ² œ "" *" " ! # 9. (a) D (0)(1) (12)(1) (22)(1) (10)(1) (5)(1) (13)(1) (11)(1) (6)(1) (2)(1) (6)(1) 87 inches ¸ ±±±± ± ± œ (b) D (12)(1) (22)(1) (10)(1) (5)(1) (13)(1) (11)(1) (6)(1) (2)(1) (6)(1) (0)(1) 87 inches ¸ ± ± ±± ± ±±±±œ 10. (a) D (1)(300) (1.2)(300) (1.7)(300) (2.0)(300) (1.8)(300) (1.6)(300) (1.4)(300) (1.2)(300) ¸ ±±±±±±± (1.0)(300) (1.8)(300) (1.5)(300) (1.2)(300) 5220 meters (NOTE: 5 minutes 300 seconds) œ (b) D (1.2)(300) (1.7)(300) (2.0)(300) (1.8)(300) (1.6)(300) (1.4)(300) (1.2)(300) (1.0)(300) ¸±±±±±±± (1.8)(300) (1.5)(300) (1.2)(300) (0)(300) 4920 meters (NOTE: 5 minutes 300 seconds) œ œ 11. (a) D (0)(10) (44)(10) (15)(10) (35)(10) (30)(10) (44)(10)
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ISM_T11_C05_A - CHAPTER 5 INTEGRATION 5.1 ESTIMATING WITH...

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