text: Calculus Volume 1
  
  Chapter 6
  
  Checkpoint
  
  6.1
  
  $1212$ units2
  
  6.2
  
  $310310$ unit2
  
  6.3
  
  $2+222+22$ units2
  
  6.4
  
  $5353$ units2
  
  6.5
  
  $5353$ units2
  
  6.7
  
  $π 2 π 2$
  
  6.8
  
  $8π8π$ units3
  
  6.9
  
  $21π21π$ units3
  
  6.10
  
  $10π310π3$ units3
  
  6.11
  
  $60π60π$ units3
  
  6.12
  
  $15π215π2$ units3
  
  6.13
  
  $8π8π$ units3
  
  6.14
  
  $12π12π$ units3
  
  6.15
  
  $11π611π6$ units3
  
  6.16
  
  $π6π6$ units3
  
  6.17
  
  Use the method of washers; $V=∫−11π[(2−x2)2−(x2)2]dxV=∫−11π[(2−x2)2−(x2)2]dx$
  
  6.18
  
  $1 6 ( 5 5 − 1 ) ≈ 1.697 1 6 ( 5 5 − 1 ) ≈ 1.697$
  
  6.19
  
  $Arc Length ≈ 3.8202 Arc Length ≈ 3.8202$
  
  6.20
  
  $Arc Length = 3.15018 Arc Length = 3.15018$
  
  6.21
  
  $π 6 ( 5 5 − 3 3 ) ≈ 3.133 π 6 ( 5 5 − 3 3 ) ≈ 3.133$
  
  6.22
  
  $12 π 12 π$
  
  6.23
  
  $70 / 3 70 / 3$
  
  6.24
  
  $24 π 24 π$
  
  6.25
  
  $88$ ft-lb
  
  6.26
  
  Approximately $43,255.243,255.2$ ft-lb
  
  6.27
  
  $156,800156,800$ N
  
  6.28
  
  Approximately 7,164,520,000 lb or 3,582,260 t
  
  6.29
  
  $M = 24 , x – = 2 5 m M = 24 , x – = 2 5 m$
  
  6.30
  
  $(−1,−1)(−1,−1)$ m
  
  6.31
  
  The centroid of the region is $(3/2,6/5).(3/2,6/5).$
  
  6.32
  
  The centroid of the region is $(1,13/5).(1,13/5).$
  
  6.33
  
  The centroid of the region is $(0,2/5).(0,2/5).$
  
  6.34
  
  $6π26π2$ units3
  
  6.35
  1. $ddxln(2x2+x)=4x+12x2+xddxln(2x2+x)=4x+12x2+x$
  2. $ddx(ln(x3))2=6ln(x3)xddx(ln(x3))2=6ln(x3)x$
  6.36
  
  $∫ x 2 x 3 + 6 d x = 1 3 ln | x 3 + 6 | + C ∫ x 2 x 3 + 6 d x = 1 3 ln | x 3 + 6 | + C$
  
  6.37
  
  $4 ln 2 4 ln 2$
  
  6.38
  1. $ddx(ex2e5x)=ex2−5x(2x−5)ddx(ex2e5x)=ex2−5x(2x−5)$
  2. $ddt(e2t)3=6e6tddt(e2t)3=6e6t$
  6.39
  
  $∫ 4 e 3 x d x = − 4 3 e −3 x + C ∫ 4 e 3 x d x = − 4 3 e −3 x + C$
  
  6.40
  1. $ddt4t4=4t4(ln4)(4t3)ddt4t4=4t4(ln4)(4t3)$
  2. $ddxlog3(x2+1)=x(ln3)(x2+1)ddxlog3(x2+1)=x(ln3)(x2+1)$
  6.41
  
  $∫ x 2 2 x 3 d x = 1 3 ln 2 2 x 3 + C ∫ x 2 2 x 3 d x = 1 3 ln 2 2 x 3 + C$
  
  6.42
  
  There are $81,377,39681,377,396$ bacteria in the population after $44$ hours. The population reaches $100100$ million bacteria after $244.12244.12$ minutes.
  
  6.43
  
  At $5%5%$ interest, she must invest $223,130.16.223,130.16.$ At $6%6%$ interest, she must invest $165,298.89.165,298.89.$
  
  6.44
  
  $38.9038.90$ months
  
  6.45
  
  The coffee is first cool enough to serve about $3.53.5$ minutes after it is poured. The coffee is too cold to serve about $77$ minutes after it is poured.
  
  6.46
  
  A total of $94.1394.13$ g of carbon remains. The artifact is approximately $13,30013,300$ years old.
  
  6.47
  1. $ddx(tanh(x2+3x))=(sech2(x2+3x))(2x+3)ddx(tanh(x2+3x))=(sech2(x2+3x))(2x+3)$
  2. $ddx(1(sinhx)2)=ddx(sinhx)−2=−2(sinhx)−3coshxddx(1(sinhx)2)=ddx(sinhx)−2=−2(sinhx)−3coshx$
  6.48
  1. $∫sinh3xcoshxdx=sinh4x4+C∫sinh3xcoshxdx=sinh4x4+C$
  2. $∫sech2(3x)dx=tanh(3x)3+C∫sech2(3x)dx=tanh(3x)3+C$
  6.49
  1. $ddx(cosh−1(3x))=39x2−1ddx(cosh−1(3x))=39x2−1$
  2. $ddx(coth−1x)3=3(coth−1x)21−x2ddx(coth−1x)3=3(coth−1x)21−x2$
  6.50
  1. $∫1x2−4dx=cosh−1(x2)+C∫1x2−4dx=cosh−1(x2)+C$
  2. $∫11−e2xdx=−sech−1(ex)+C∫11−e2xdx=−sech−1(ex)+C$
  6.51
  
  $52.95 ft 52.95 ft$
  
  Section 6.1 Exercises
  
  1.
  
  $32 3 32 3$
  
  3.
  
  $13 12 13 12$
  
  5.
  
  $36 36$
  
  7.
  
  243 square units
  
  9.
  
  4
  
  11.
  
  $2(e−1)2e2(e−1)2e$
  
  13.
  
  $1313$
  
  15.
  
  $343343$
  
  17.
  
  $5252$
  
  19.
  
  $1212$
  
  21.
  
  $9292$
  
  23.
  
  $9292$
  
  25.
  
  $332332$
  
  27.
  
  $e−2e−2$
  
  29.
  
  $274274$
  
  31.
  
  $43−ln(3)43−ln(3)$
  
  33.
  
  $1212$
  
  35.
  
  $1212$
  
  37.
  
  $−2(2−π)−2(2−π)$
  
  39.
  
  $1.067 1.067$
  
  41.
  
  $0.852 0.852$
  
  43.
  
  $7.523 7.523$
  
  45.
  
  $3 π − 4 12 3 π − 4 12$
  
  47.
  
  $1.429 1.429$
  
  49.
  
  $33,333.3333,333.33$ total profit for $200200$ cell phones sold
  
  51.
  
  $3.2633.263$ mi represents how far ahead the hare is from the tortoise
  
  53.
  
  $343 24 343 24$
  
  55.
  
  $4 3 4 3$
  
  57.
  
  $π − 32 25 π − 32 25$
  
  Section 6.2 Exercises
  
  63.
  
  8 units3
  
  65.
  
  $32323232$ units3
  
  67.
  
  $724πr2h724πr2h$ units3
  
  69.
  
  $π24π24$ units3
  
  71.
  
  $22$ units3
  
  73.
  
  $π240π240$ units3
  
  75.
  
  $4096π54096π5$ units3
  
  77.
  
  $8π98π9$ units3
  
  79.
  
  $π2π2$ units3
  
  81.
  
  $207π207π$ units3
  
  83.
  
  $4π54π5$ units3
  
  85.
  
  $16π316π3$ units3
  
  87.
  
  $ππ$ units3
  
  89.
  
  $16π316π3$ units3
  
  91.
  
  $72π572π5$ units3
  
  93.
  
  $108π5108π5$ units3
  
  95.
  
  $3π103π10$ units3
  
  97.
  
  $26π26π$ units3
  
  99.
  
  $9π9π$ units3
  
  101.
  
  $π20(75−4ln5(2))π20(75−4ln5(2))$ units3
  
  103.
  
  $m2π3(b3−a3)m2π3(b3−a3)$ units3
  
  105.
  
  $4a2bπ34a2bπ3$ units3
  
  107.
  
  $2π22π2$ units3
  
  109.
  
  $2ab2π32ab2π3$ units3
  
  111.
  
  $V=2π3(2r3-3r2h+h3)V=2π3(2r3-3r2h+h3)$ units3
  
  113.
  
  $π3(h+R)(h−2R)2π3(h+R)(h−2R)2$ units3
  
  Section 6.3 Exercises
  
  115.
  
  $54π54π$ units3
  
  117.
  
  $81π81π$ units3
  
  119.
  
  $512π7512π7$ units3
  
  121.
  
  $2π2π$ units3
  
  123.
  
  $2π32π3$ units3
  
  125.
  
  $2π2π$ units3
  
  127.
  
  $4π54π5$ units3
  
  129.
  
  $64π364π3$ units3
  
  131.
  
  $32π532π5$ units3
  
  133.
  
  $7 π 6 7 π 6$
  
  135.
  
  $48 π 48 π$
  
  137.
  
  $113 π 5 113 π 5$
  
  139.
  
  $512 π 7 512 π 7$
  
  141.
  
  $96π596π5$ units3
  
  143.
  
  $28π1528π15$ units3
  
  145.
  
  $3π103π10$ units3
  
  147.
  
  $π6·22/35-1110=π1012·22/3-11≈2.5286π6·22/35-1110=π1012·22/3-11≈2.5286$ units3
  
  149.
  
  $0.98760.9876$ units3
  
  151.
  
  $3232$ units3
  
  153.
  
  $496π15496π15$ units3
  
  155.
  
  $398π15398π15$ units3
  
  157.
  
  $15.907415.9074$ units3
  
  159.
  
  $13πr2h13πr2h$ units3
  
  161.
  
  $πr2hπr2h$ units3
  
  163.
  
  $πa2πa2$ units3
  
  Section 6.4 Exercises
  
  165.
  
  $2 26 2 26$
  
  167.
  
  $2 17 2 17$
  
  169.
  
  $π 6 ( 17 17 − 5 5 ) π 6 ( 17 17 − 5 5 )$
  
  171.
  
  $13 13 − 8 27 13 13 − 8 27$
  
  173.
  
  $4 3 4 3$
  
  175.
  
  $2.0035 2.0035$
  
  177.
  
  $123 32 123 32$
  
  179.
  
  $10 10$
  
  181.
  
  $20 3 20 3$
  
  183.
  
  $1 675 ( 229 229 − 8 ) 1 675 ( 229 229 − 8 )$
  
  185.
  
  $1 8 ( 4 5 + ln ( 9 + 4 5 ) ) 1 8 ( 4 5 + ln ( 9 + 4 5 ) )$
  
  187.
  
  $1.201 1.201$
  
  189.
  
  $15.2341 15.2341$
  
  191.
  
  $49 π 3 49 π 3$
  
  193.
  
  $70 π 2 70 π 2$
  
  195.
  
  $8 π 8 π$
  
  197.
  
  $120 π 26 120 π 26$
  
  199.
  
  $π 6 ( 17 17 − 1 ) π 6 ( 17 17 − 1 )$
  
  201.
  
  $9 2 π 9 2 π$
  
  203.
  
  $10 10 π 27 ( 73 73 − 1 ) 10 10 π 27 ( 73 73 − 1 )$
  
  205.
  
  $25.645 25.645$
  
  207.
  
  $π ( π + 2 ) π ( π + 2 )$
  
  209.
  
  $10.5017 10.5017$
  
  211.
  
  $2323$ ft
  
  213.
  
  $2 2$
  
  215.
  
  217.
  
  Section 6.5 Exercises
  
  219.
  
  $150150$ ft-lb
  
  221.
  
  $200 J 200 J$
  
  223.
  
  $11$ J
  
  225.
  
  $39 2 39 2$
  
  227.
  
  $ln ( 243 ) ln ( 243 )$
  
  229.
  
  $332 π 15 332 π 15$
  
  231.
  
  $100 π 100 π$
  
  233.
  
  $20 π 15 20 π 15$
  
  235.
  
  $66$ J
  
  237.
  
  $55$ cm
  
  239.
  
  $3636$ J
  
  241.
  
  $18,75018,750$ ft-lb
  
  243.
  
  Weight $=323×109lb=323×109lb$
  
  Work $=43×1012ft-lb=43×1012ft-lb$
  
  245.
  
  $9.71 × 10 2 N m 9.71 × 10 2 N m$
  
  247.
  
  a. $3,000,0003,000,000$ lb, b. $750,000750,000$ lbs
  
  249.
  
  $23.25π23.25π$ million ft-lb
  
  251.
  
  $A ρ H 2 2 A ρ H 2 2$
  
  253.
  
  Section 6.6 Exercises
  
  255.
  
  $5 4 5 4$
  
  257.
  
  $( 2 3 , 2 3 ) ( 2 3 , 2 3 )$
  
  259.
  
  $( 7 4 , 3 2 ) ( 7 4 , 3 2 )$
  
  261.
  
  $3 L 4 3 L 4$
  
  263.
  
  $π 2 π 2$
  
  265.
  
  $e 2 + 1 e 2 − 1 e 2 + 1 e 2 − 1$
  
  267.
  
  $π 2 − 4 π π 2 − 4 π$
  
  269.
  
  $1 4 ( 1 + e 2 ) 1 4 ( 1 + e 2 )$
  
  271.
  
  $( a 3 , b 3 ) ( a 3 , b 3 )$
  
  273.
  
  $( 0 , π 8 ) ( 0 , π 8 )$
  
  275.
  
  $( 0 , 3 ) ( 0 , 3 )$
  
  277.
  
  $( 0 , 4 π ) ( 0 , 4 π )$
  
  279.
  
  $( 5 8 , 1 3 ) ( 5 8 , 1 3 )$
  
  281.
  
  $2 m π 3 2 m π 3$
  
  283.
  
  $π a 2 b π a 2 b$
  
  285.
  
  $( 4 3 π , 4 3 π ) ( 4 3 π , 4 3 π )$
  
  287.
  
  $( 1 2 , 2 5 ) ( 1 2 , 2 5 )$
  
  289.
  
  $( 0 , 28 9 π ) ( 0 , 28 9 π )$
  
  291.
  
  Center of mass: $(a6,4a25),(a6,4a25),$ volume: $2πa492πa49$
  
  293.
  
  Volume: $2π2a2(b+a)2π2a2(b+a)$
  
  Section 6.7 Exercises
  
  295.
  
  $1 x 1 x$
  
  297.
  
  $− 1 x ( ln x ) 2 − 1 x ( ln x ) 2$
  
  299.
  
  $ln ( x + 1 ) + C ln ( x + 1 ) + C$
  
  301.
  
  $ln ( x ) + 1 ln ( x ) + 1$
  
  303.
  
  $cot ( x ) cot ( x )$
  
  305.
  
  $7 x 7 x$
  
  307.
  
  $csc ( x ) sec x csc ( x ) sec x$
  
  309.
  
  $−2 tan x −2 tan x$
  
  311.
  
  $1 2 ln ( 5 3 ) 1 2 ln ( 5 3 )$
  
  313.
  
  $2 − 1 2 ln ( 5 ) 2 − 1 2 ln ( 5 )$
  
  315.
  
  $1 ln ( 2 ) − 1 1 ln ( 2 ) − 1$
  
  317.
  
  $1 2 ln ( 2 ) 1 2 ln ( 2 )$
  
  319.
  
  $1 3 ( ln x ) 3 1 3 ( ln x ) 3$
  
  321.
  
  $2 x 3 x 2 + 1 x 2 − 1 2 x 3 x 2 + 1 x 2 − 1$
  
  323.
  
  $x −2 − ( 1 / x ) ( ln x − 1 ) x −2 − ( 1 / x ) ( ln x − 1 )$
  
  325.
  
  $e x e − 1 e x e − 1$
  
  327.
  
  $1 1$
  
  329.
  
  $− 1 x 2 − 1 x 2$
  
  331.
  
  $π − ln ( 2 ) π − ln ( 2 )$
  
  333.
  
  $1 x 1 x$
  
  335.
  
  $e 5 − 6 units 2 e 5 − 6 units 2$
  
  337.
  
  $ln ( 4 ) − 1 units 2 ln ( 4 ) − 1 units 2$
  
  339.
  
  $2.8656 2.8656$
  
  341.
  
  $3.1502 3.1502$
  
  Section 6.8 Exercises
  
  349.
  
  True
  
  351.
  
  False; $k=ln(2)tk=ln(2)t$
  
  353.
  
  $2020$ hours
  
  355.
  
  No. The relic is approximately $871871$ years old.
  
  357.
  
  $71.9271.92$ years
  
  359.
  
  $55$ days $66$ hours $2727$ minutes
  
  361.
  
  $12 12$
  
  363.
  
  $8.618 % 8.618 %$
  
  365.
  
  $6766.76 6766.76$
  
  367.
  
  $99$ hours $1313$ minutes
  
  369.
  
  $239,179239,179$ years
  
  371.
  
  $P′(t)=43e0.01604t.P′(t)=43e0.01604t.$ The population is always increasing.
  
  373.
  
  The population reaches $1010$ billion people in $2027.2027.$
  
  375.
  
  $P′(t)=2.259e0.06407t.P′(t)=2.259e0.06407t.$ The population is always increasing.
  
  Section 6.9 Exercises
  
  377.
  
  $e x and e − x e x and e − x$
  
  379.
  
  381.
  
  383.
  
  385.
  
  $3 sinh ( 3 x + 1 ) 3 sinh ( 3 x + 1 )$
  
  387.
  
  $− tanh ( x ) sech ( x ) − tanh ( x ) sech ( x )$
  
  389.
  
  $4 cosh ( x ) sinh ( x ) 4 cosh ( x ) sinh ( x )$
  
  391.
  
  $x sech 2 ( x 2 + 1 ) x 2 + 1 x sech 2 ( x 2 + 1 ) x 2 + 1$
  
  393.
  
  $6 sinh 5 ( x ) cosh ( x ) 6 sinh 5 ( x ) cosh ( x )$
  
  395.
  
  $1 2 sinh ( 2 x + 1 ) + C 1 2 sinh ( 2 x + 1 ) + C$
  
  397.
  
  $1 2 sinh 2 ( x 2 ) + C 1 2 sinh 2 ( x 2 ) + C$
  
  399.
  
  $1 3 cosh 3 ( x ) + C 1 3 cosh 3 ( x ) + C$
  
  401.
  
  $ln ( 1 + cosh ( x ) ) + C ln ( 1 + cosh ( x ) ) + C$
  
  403.
  
  $cosh ( x ) + sinh ( x ) + C cosh ( x ) + sinh ( x ) + C$
  
  405.
  
  $4 1 − 16 x 2 4 1 − 16 x 2$
  
  407.
  
  $sinh ( x ) cosh 2 ( x ) + 1 sinh ( x ) cosh 2 ( x ) + 1$
  
  409.
  
  $− csc ( x ) − csc ( x )$
  
  411.
  
  $− 1 ( x 2 − 1 ) tanh −1 ( x ) − 1 ( x 2 − 1 ) tanh −1 ( x )$
  
  413.
  
  $1 a tanh −1 ( x a ) + C 1 a tanh −1 ( x a ) + C$
  
  415.
  
  $x 2 + 1 + C x 2 + 1 + C$
  
  417.
  
  $cosh −1 ( e x ) + C cosh −1 ( e x ) + C$
  
  419.
  
  421.
  
  $37.30 37.30$
  
  423.
  
  $y = 1 c cosh ( c x ) y = 1 c cosh ( c x )$
  
  425.
  
  $−0.521095 −0.521095$
  
  427.
  
  $10 10$
  
  Review Exercises
  
  435.
  
  False
  
  437.
  
  False
  
  439.
  
  441.
  
  $162 π 5 162 π 5$
  
  443.
  
  a. $4,4,$ b. $128π7,128π7,$ c. $64π564π5$
  
  445.
  
  a. $1.949,1.949,$ b. $21.952,21.952,$ c. $17.09917.099$
  
  447.
  
  a. $316,316,$ b. $452π15,452π15,$ c. $31π631π6$
  
  449.
  
  $245.282 245.282$
  
  451.
  
  Mass: $12,12,$ center of mass: $(1835,922)(1835,922)$
  
  453.
  
  $17 + 1 8 ln ( 33 + 8 17 ) 17 + 1 8 ln ( 33 + 8 17 )$
  
  455.
  
  Volume: $3π4,3π4,$ surface area: $π(2−sinh−1(1)+sinh−1(16)−25716)π(2−sinh−1(1)+sinh−1(16)−25716)$
  
  457.
  
  11:02 a.m.
  
  459.
  
  $π ( 1 + sinh ( 1 ) cosh ( 1 ) ) π ( 1 + sinh ( 1 ) cosh ( 1 ) )$
  
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