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Calculus12-Integrals
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Contents 1ApproximatingAreaUnderCurves 4 RiemannSums ..................................... 6 InfiniteSums(Series) 7 TrapezoidalRule 8 PracticeExercises 10 2AntiderivativesandIndefiniteIntegrals 12 TheIndefiniteIntegral ................................. 14 PracticeExercises 15 3TheDefiniteIntegral 17 Average(Mean)Value 18 IntegralasaFunction 19 PracticeExercises .................................... 20 4TheFundamentalTheoremofCalculus 22 DifferentialEquationswithInitialConditions 24 PracticeExercises 25 5ProblemsInvolvingMotion 27 DisplacementvsDistance 28 PracticeExercises .................................... 30 6TheSubstitutionRule 32 PracticeExercises 35 7AnswerKey 38 Section1-Exercises 38 Section2-Exercises 38 Section3-Exercises .................................. 39 Section4-Exercises .................................. 40 Section5-Exercises 40 Section6-Exercises 41
1ApproximatingAreaUnderCurves
Findingtheareaunderacurvecanvaryindifficultydependingonthefunction.Linear functionsareeasybecausetheytakeonthebasicshapesoftriangles/rectangles.
Example1.1 Findtheareaunderthecurvebelowfrom x =0to x =8 x y
Circlesorpartsofcirclesarealsosolvablebytakingtheareaofacircle.
Example1.2 Findtheareaunderthecurvebelowfrom x = 2to x =2
Forotherfunctionsthatdon’ttakeonaregulargeometricshape,itcanbedifficultto calculatetheareaunderthecurve.
Forexample,howwouldwebegincalculatingtheareaunderneaththecurvebelow?
Integrals —4
x y
x
y
Integrals —5 Oneapproachistobeginbytakingapproximationsusingshapesweknowtheareasof.For example,rectanglesandtrapezoids.Theaccuracydependsonhowmanyrectangles/trapezoids areused.Ifthenumberofrectangles, n,istakentoaninfiniteamount,thentheapproximatedareawillbeinfinitelyclosetotheactualarea.
Example1.3 Thegraphbelowleftshowsthepreviouscurveanduses8rectanglestoapproximatetheareaunderneaththecurve.Belowright,uses16rectangles.
Example1.4 Draw5rectanglestoshowtheapproximatedareaunderthecurve y = √x ontheinterval4 ≤ x ≤ 9.Noneedtocalculate.
(a) Usingleftendpoints
(b) Usingrightendpoints
(c) Usingmidpoints
x y x y
(a) x y (b) x y (c) x y
RiemannSums
Wedefinetheprocessofusingafinitenumberofrectanglestoapproximatetheareaunderneaththecurveasthe RiemannSum
Approximatingtheleftendpointsiscommonlyreferredtoas LRAM (LeftRectangular ApproximationMethod),orLeftRiemannSum.
Similarly, RRAM forRightRectangularApproximationMethod,orRightRiemannSum.
And, MRAM forMidpointRectangularApproximationMethod,orMidpointRiemann Sum.
Example1.5 Approximatetheareaof f (x)= x2 ontheinterval[0, 2]using
(a) LRAM4
(b) RRAM4
(c) MRAM4
Thesubscriptdenoteshowmanyrectanglestoapproximatewith.
Integrals —6
InfiniteSums(Series)
Wecancalculateinfinitesumsusingsomewell-knownformulasforspecificseries.
TheDefiniteIntegralofaContinuousFunctionon [a,b]
Let f becontinuouson[a,b],andlet[a,b]bepartitionedinto n subintervalsofequal length,∆x = (b a) n .Thenthedefiniteintegralof f over[a,b]isgivenby
whereeach ck ischosenarbitrarilyinthe kth subinterval.
Example1.6 Findtheareaunder f (x)= x2 from x =0to x =2,usingtheabove definition.
Let∆x = 2 0 n = 2 n .Wecanchoose c
(RRAM).
Integrals —7
n i=1 1= n n i=1 i = n(
2 n i=1 i2 = n(n +1)(2n +1) 6 n i=1 i3 = n2
n +1)
n +1)
(
2 4
lim n→∞ n k=1 f (ck
)∆x
k
2k
A =lim n→∞ n k=1 f (ck )∆x =lim n→∞ n k=1 2k n 2 2 n =lim n→∞ 8 n3 · n k=1 k2 =lim n→∞ 8 n3 n(n +1)(2n +1) 6 =lim n→∞ 8 n3 2n3 +3n2 + n 6 =lim n→∞ 16n3 +24n2 +8n 6n3 =lim n→∞ 8 3 + 4 n + 4 3n2
=
n
= 8 3
TrapezoidalRule
Usingasmallnumberofrectanglestoapproximatetheareaunderacurveisnotoftenvery accurate.Insteadofusingrectangles,usingtrapezoidsprovidesamoreaccurateapproximationoftheareaunderacurve.
Recalltheareaofatrapezoidis
Example1.7 Belowisavisualrepresentationofusing4trapezoidsonthecurve y = x2 on theinterval[0, 2]
The trapezoidalrule is
where[a,b]ispartitionedinto n subintervalsofequallength, h = (b a) n .Equivalently,
Integrals —8
A
(b1 + b2) 2 h
=
x y
T = h 2 (y0 +2y1 +2y2 + ... +2yn 1 + yn)
T
= LRAMn + RRAMn 2
Example1.8 For y = ex on[0, 2],calculatethefollowinganddetermineifthemethodused isanunder-approximationoranover-approximation.Usetheprovidedsketchtohelp.
Integrals —9
x
y y = ex
(a) L4
(b) R4
(c) M4
(d) T4
1. Given f (x)=2x +3overtheinterval[0, 4].Calculate
(a) R2 (b) R4 (c) Theexactareabyanalyzingthesketchgeometrically.
2. Sketchtheregionbelow f (x)=1 x3 overtheinterval[0, 1].Showrectangularapproximations L5,R5, and M5.Explainwhichapproximationisalowerbound,whichisan upperbound,andwhichisthebestapproximationoftheareaoftheregion.
3. Calculatethefollowingrectangularapproximations.
(a) R4 for f (x)= x2 over[0, 1] (b) M4 for f (x)=1 x2 over[0, 1]
Integrals —10
PracticeExercises
(c) L5 for f (x)=2x over[0, 5]
(d) M3 for f (x)= √x 1over[1, 4]
4. Giventhetableofvaluesbelow,calculatethefollowing
L5
(b) R5
T5
Integrals —11
x
0 0.4 0.8 1.2 1.6 2.0 f (x) 20 18 15 19 18 14
(a)
(c)
2AntiderivativesandIndefiniteIntegrals
Afterthestudyofderivatives,wecanbegintoworkwithantiderivativeswhichwillleadinto Integrals.
Wetypicallyuse F (x)todenotetheantiderivativeof f (x).Thatis
F ′(x)= f (x)
Notethatthederivativeofanyconstant, C,isequaltozero.So d dx F (x)+ C = f (x)
Example2.1 If f (x)=2x,find F (x),whichisthe general antiderivativeto f (x).
TableofGeneralAntiderivatives
Function f (x)
f (x)= k
f (x)= xn where n = 1
f (x)= 1 x
f (x)= ekx
f (x)=cos(kx)
f (x)=sin(kx)
Antiderivativeof f (x)
F (x)= kx + C
F (x)= 1 n +1 xn+1 + C
F (x)=ln |x| + C
F (x)= 1 k ekx + C
F (x)= 1 k sin(kx)+ C
F (x)= 1 k cos(kx)+ C
Integrals —12
Example2.2 Findthegeneralantiderivativetothefollowing
f (x)=2x2 x +7
f (x)=cos x sin x
f (x)= 3e x +6e2x
(e) f (x)= √x + 1 x
f (x)= 2 x2 5 x + x
(f) f (x)=2sin x cos x
f (x)= 3csc2x cot2x
f (x)=76x
—13
Integrals
(a)
(b)
(c)
(d)
(g)
(h)
TheIndefiniteIntegral
TheIntegralistheinverseoftheDerivative.Wecan integrate afunction,orfindthe integration ofafunction.
Recallthattheactionoftakingthederivativeofafunctionwithrespectto x is
Thentheinverseofthederivativeistheintegral.
Theindefiniteintegralof f (x)is always the generalantiderivative of f (x).
=
where C isanyconstant. The dx is required toindicatewhichvariabletointegratewithrespectto.
Example2.3 Findthefollowing
Integrals —14
d dx f (x)= f ′(x
)
f ′(x
dx
f (x
)
=
)+ C
Thatis 2
butinstead 2
xdx = x 2
xdx
x 2 + C
(b)
(c) (x 4 3x 3 + √x) dx (d) (cos(5x) sec 2 (3x)) dx
(a) 2sin xdx
1 x dx
1. Findthegeneralantiderivativeof
(a) f (x)=2x +1
(b) f (x)=4x3 11
(c) f (x)=16x9 9x4 +3x
(d) f (x)= √x + 3 √x
(e) f (x)= 2 x7 + x5 2
(f) f (x)= 2 x3 3 x2
(g) f (x)= 1 x
(h) f (x)=sin x +cos x
(i) f (x)=sin2x +2cos x (j) f (x)= 3cos5x +8sin x
(k) f (x)= 4cos(x +2) (l) f (x)= ex + e x
Integrals —15
PracticeExercises
2. Findthefollowingintegrals
(a) 5 dx
(b) 4xdx
(c) cos xdx
(d) (3x 2 4x +7) dx
(e) (8s 3 12s 2 +5) ds
(f) y 4/3 dy
(g) 4 x2 dx (h) 1 t√t dt
(i) sec 2 θdθ (j) 1 x dx
Integrals —16
3TheDefiniteIntegral
Theintegralofafunctioncanbeconsideredtheareaunderneaththefunctionbutabovethe x-axis.Anintegralcanbenegativeiftheareaisbelowthe x-axis.
Thenotationusedtorepresentthe definiteintegral from[a,b]of f (x)is
DefiniteIntegralRules
1.
Example3.1 Findeachofthefollowingintegrals,given
Integrals —17
b a f (x
dx
)
x
dx
a b f (x
dx
f
x
dx =0
b a kf (x) dx = k b a f (x) dx 4. b a f (x) ± g(x) dx = b a f (x) dx ± b a g(x) dx 5. b a
(x
dx
c
f
x
dx = c a f
x
dx
b a f (
)
=
)
2. a a
(
)
3.
f
)
+
b
(
)
(
)
1 1 f (x) dx =5, 4 1 f (x) dx = 2, 1 1 h(x) dx =7 (a) 1 4 f (x) dx (b) 4 1 f (x) dx (c) 1 1 [2f (x)+3h(x)] dx
Example3.2 Useintegralstorepresentthepartsinthegraphbelow
Average(Mean)Value
If f isintegrableon[a,b],its average(mean)value on[a,b]is
Theaverage(mean)valueis not arateofchange,butanaverage y-value.
Example3.3 If 10 2 f (x) dx =80,finditsaveragevalue.
Integrals —18
x y A1 A2 y =3x x2 3 5 A1 = A2 = TotalArea=
b a b a f (x
av(f )= 1
) dx
IntegralasaFunction
Theboundsofanintegralcanalsoincludeavariable.Thismakestheintegralitselfa function. g(x)= x 0 f (t) dt
Notethedifferentvariable, t,usedinthefunction.Thisisjusttopreventconfusionfrom g(x),andeithervariableisinterchangeable.Othervariablescanbeusedaswell.
Twomaintypesofquestionsusethisnotation.Oneisunderstandingthegraphicalnature ofthis.Theotheristreatingitasafunctionandderiving(moreonthislater).
Theabove g(x)canbeinterpretedastheaccumulationofarea(positiveand/ornegative) underthecurveof y = f (x)from0to x.
Example3.4 Let g(x)= x 3 f (t) dt forthegraphof y = f (x)below.
y y = f (x)
Integrals —19
x
(a) Find g( 3)
(b) Find g(1)
(c) Find g(3)
1. Sketchthegraphofthefollowingtoevaluatetheintegral.
2. Giventhefollowingintegralvalues
Integrals —20
PracticeExercises
(a) 4 1 (2x +3) dx (b) 4 1 (3x 2) dx (c) 5 0 4xdx (d) 7 3 (x 4) dx
11 6 f (x) dx = 7 11 6 g(x) dx =24 determinethevaluesof (a) 6 11 9f (x) dx (b) 11 6 (6g(x) 10f (x)) dx
3. Determinethevalueof 9 2 f (x) dx giventhat 2 5 f (x) dx =3and 9 5 f (x) dx =8.
4. Determinethevalueof 20 4 f (x) dx giventhat 0 4 f (x) dx = 2, 0 31 f (x) dx =19and 31 20 f (x) dx =8.
5. Giventhegraphof f (x)belowandlet g(x)= x 0 f (t) dt.Evaluatethefollowing.
Integrals —21
x y y = f (x) 1 2 3 4 5 6 7 8 9 10 11 12 3 2 1 1 2 3
(a) g(2)
(b) g(3)
(c) g(8)
(d) g(12)
4TheFundamentalTheoremofCalculus
TheFundamentalTheoremofCalculus If f iscontinuousateverypointof[a,b],andif F isanyantiderivativeof f on[a,b], then
Example4.2 Evaluate
Integrals —22
b a f (x) dx = F (b) F (a
)
5 2 (2x 3 3x 2 +7x +2) dx
Example4.1 Find
8 1 1 3 √x2 dx Example4.3 Find 4 1 t2 + √t 2 t dt
The FundamentalTheoremofCalculus givestheexact“area”underneaththecurveon anygiveninterval.However,itdoesincludenegative“areas”inthetotal,soanindefinite integralmaybepositive,negative,orzero.
Example4.4 Find
Example4.5 Find
Integrals —23
π 0
x y
2
sin xdx
2 2
3
x
dx
DifferentialEquationswithInitialConditions
A differentialequation isanequationwherethederivativeofafunctionisgiven.This meansweknowthemodelofthefunction’srateofchange. Normally,whenonlythederivativeisknown,wecanonlyfindthegeneralantiderivative. However,ifaninitialconditionwasgiven,thenwecansolvefortheconstant.
Example4.6 Solvefor s,if ds dt =2t,withtheinitialcondition: s =3when t =0.
Example4.7 Findthecurve y = F (x)thatpassesthrough( 1, 0)andsatisfies dy dx =6x2 +6x
Integrals —24
1. Evaluatethefollowingdefiniteintegrals.
Integrals —25
PracticeExercises
(a) 3 1 (2x 2 12x +13) dx (b) 3 0 ( x 3 +3x 2 2) dx (c) 0 1 (x 5 4x 3 +4x +4) dx (d) 0 3 (4 3 √x) dx (e) 1 4 4 x3 dx (f) 1 3 4 x dx (g) π 6 π 4 2cos xdx (h) 1 1 e 2x 2 dx
2. Findtheparticularsolutionforthedifferentialequationthatsatisfiesthegiveninitial condition.
(a) dy dx = 4x +1 y( 1)= 3 (b) dy dx = 2sin xy( 2π 3 )=0
(c) dy dx = e3x y(0)=4
(d) dy dx = 2 3x y(e3)=4
Integrals —26
5ProblemsInvolvingMotion
Recallthattheaccelerationfunction a(t)isthederivativeofthevelocityfunction v(t),which inturn,isthederivativeofthepositionfunction s(t).
Example5.1 Anobjectmovesinastraightlinewithvelocity v =6t 3t2,where v is measuredinmetrespersecond.
(a) Whenistheobjectnotmoving?
(b) Howfardoestheobjectmoveinthefirstsecond?
(c) Howfardoestheobjectmoveinthefirsttwoseconds?
(d) Theobjectisbackwhereitstartedwhen t =3.Howfardidittraveltogetthere?
Integrals —27
DisplacementvsDistance
Ingeneral,theareaunderneatharatecurveisthenetchange.Forexample,thearea underneathavelocitycurveisthenetchangeinposition,or displacement
Reminder: Areabelowthe x-axisisconsiderednegative.
Displacement
Thechangeinpositionfromavelocityfunction v(t)from a ≤ t ≤ b is
a v(t) dt
Distanceanddisplacementarenotthesamething.Sometimesdistanceisequaltodisplacement,butnotalways.
Distancecanbeconsidered totalarea ofthecurve,factoringinbothpositiveandnegative areas.
Distance
Thechangeinpositionfromavelocityfunction v(t)from a ≤ t ≤ b is b a |v(t)| dt
Example5.2 Thevelocityfunction(measuredinmetrespersecond)foraparticlemoving alongastraightlineisgivenby v(t)=4 t.Findthedisplacementanddistancetravelled overthetimeinterval0 ≤ t ≤ 4.
Integrals —28
b
Example5.3 Thevelocityfunction(measuredinmetrespersecond)foraparticlemoving alongastraightlineisgivenby v(t)=2t 4.Findthedisplacementanddistancetravelled overthetimeinterval0 ≤ t ≤ 4.
Example5.4 Thevelocityfunction(measuredinmetrespersecond)foraparticlemoving alongastraightlineisgivenby v(t)= t2 9.Findthedisplacementanddistancetravelled overthetimeinterval0 ≤ t ≤ 4.
Example5.5 Astoneistossedupwardwithavelocityof8m/sfromtheedgeofacliff63 mhigh.Howlongwillittakethestonetohitthegroundatthefootofthecliff?Gravity’s accelerationis9.8m/s2 downwards.
Integrals
—29
PracticeExercises
1. Foreachfunction v(t)isthevelocityinm/secofaparticlemovingalongthe x-axis. Find
i. Theparticle’sdisplacementforthegiventimeinterval.
ii. If s(0)=3,whatistheparticle’sfinalposition?
iii. Findthetotaldistancetraveledbytheparticle.
(a) v(t)=5cos t, 0 ≤ t ≤ 2π
(b) v(t)=6sin3t, 0 ≤ t ≤ π 2
(c) v(t)=49 9.8t, 0 ≤ t ≤ 10
(d) v(t)=6t2 18t +12, 0 ≤ t ≤ 2
Integrals —30
2. Acaracceleratesfromrestat(1+3√t)mph/secfor9seconds.
(a) Whatisthevelocityafter9seconds?
(b) Howfardoesittravelinthose9seconds?
3. AccelerationduetoEarth’sgravityis32ft/sec2.Fromgroundlevel,aprojectileisfired straightupwardwithavelocityof90feetpersecond.
(a) Whatisthevelocityafter3seconds?
(b) Whendoesithittheground?
(c) Whenithitstheground,whatisthetotaldistanceithastraveled?
Integrals —31
6TheSubstitutionRule
Insomeintegrals,itisdifficulttoseeanobviousantiderivative,especiallywhenthereseems tobemorethanonefunctionbeingmultipliedordivided.
Recallthechainrule.
Example6.1 Findthederivativeof y =sin2 x,thenfind (2sin x cos x) dx
SubstitutionRuleforIndefiniteIntegrals
If u = g(x),then
Example6.2 Find (2sin x cos x) dx bymakingthesubstitution u =sin x
Integrals —32
f (g(x))g ′(x) dx = f (u
) du
Example6.3 Find (x 2 5)2xdx bymakingthesubstitution u = x2 5
Example6.4 Find x2 √1 x3 dx bymakingthesubstitution u =1 x3
Example6.5 Find ln x x dx
Integrals —33
Example6.6 Find (2+sin x)10 cos xdx
SubstitutionRuleforDefiniteIntegrals
If u = g(x),then b a f (g(x))g ′(x) dx = g(b) g(a) f (u) du
Example6.7 Evaluate 9 1 e√x √x dx
Integrals —34
1. Evaluatethefollowingindefiniteintegrals. (a) 9x 2( 3x 3 +1)3 dx (b) 12x 3(3x 4 +4)4 dx (c) ( 2x 4 4)4 ·−32x 3 dx
(e 4x 4) 1 5 · 8e 4x dx
12x 2( 4x 3 +2) 3 dx
(3x 5 3) 3 5 · 5x 4 dx
Integrals —35
PracticeExercises
(e)
(f)
(d)
(g) sin(5x 2 3) · 20xdx
(h) 15x 3 · sec 2 (4x 4 2) dx
(i) 5cos( 4+ln4x) x dx
(j) 36x 3 sec(3x 4 +3) tan(3x 4 +3) dx
(k) 12x2 x3 +2 dx (l) 20e5x e5x +3 dx
Integrals —36
2. Evaluateeachdefiniteintegral
3. Expresseachdefiniteintegralintermsof u,butdonotevaluate.
Integrals —37
(a) 2 0 x√4 x2 dx (b) 1 0 x 4(x 5 +1)5 dx (c) 1 1/2 1+ 1 x 5 x2 dx (d) 2 1 (x +1)e 3x2 +6x 4 dx
(a) 0 1 8x (4x2 +1)2 dx; u =4x 2 +1 (b) 1 0 12x 2(4x 3 1)3 dx; u =4x 3 1
7AnswerKey
Section1-Exercises
1. (a) R2 =36
(b) R4 =32
(c) 28
2. L5:Upperbound, R5:Lowerbound, M5:BestApproximation
3. (a) 15 32
(b) 0.671875
(c) 31
(d) ≈ 3.513
4. (a) L5 =36
(b) R5 =33 6
(c) T5 =34.8
Section2-Exercises
1. (a) F (x)= x2 + x + C
(b) F (x)= x4 11x + C
(c) F (x)= 8 5 x10 9 5 x5 + 3 2 x2 + C
(d) F (x)= 2 3 x 3 2 + 3 4 x 4 3 + C
(e) F (x)= 1 3 x 6 + 1 12 x6 + C
(f) F (x)= x 2 +3x 1 + C
(g) F (x)=ln |x| + C
(h) F (x)= cos x +sin x + C
(i) F (x)= 1 2 cos2x +2sin x + C
(j) F (x)= 3 5 sin5x 8cos x + C
(k) F (x)= 4sin(x +2)+ C
(l) F (x)= ex e x + C
Integrals —38
2. (a) 5x + C
(b) 2x2 + C
(c) sin x + C
(d) x3 2x2 +7x + C
(e) 2s4 4s3 +5s + C
(f) 3y 1 3 + C
(g) 4 x + C
(h) 2 √t + C
(i) tan θ + C
(j) ln |x| + C
Section3-Exercises
1. (a) 24
(b) 16.5
(c) -50
(d) 4
2. (a) 63
(b) 214
3. 5
4. -29
5. (a) 4 5
(b) 3
(c) 7 5
(d) 7 5+2π
—39
Integrals
Section4-Exercises
1. (a) 14 3
(b) 3 4
(c) 17 6
(d) 9 3 √3
(e) 15 8
(f) 4ln(3)
(g) 1+ √2
(h) 1 2 1 2e4
2. (a) y = 2x2 + x
(b) y = 2cos x 1
(c) y = 1 3 e3x + 11 3
(d) y = 2 3 ln |x| +2
Section5-Exercises
1. (a) i).0mii).3miii).20m
(b) i).2mii).5miii).6m
(c) i).0mii).3miii).245m
(d) i).4mii).7miii).6m
2. (a) 63mph
(b) 234.9miles
3. (a) 6ft/sec
(b) 5.625seconds
(c) 253 125ft
Integrals —40
Section6-Exercises
1. (a) 1 4 ( 3x3 +1)4 + C
(b) 1 5 (3x4 +4)5 + C
(c) 4 5 ( 2x4 4)5 + C
(d) 5 3 (e4x 4) 6 5 + C
(e) 1 2( 4x3 +2) 2 + C
(f) 5 24 (3x5 3) 8 5 + C
(g) 2cos(5x2 3)+ C
(h) 15 16 tan(4x4 2)+ C
(i) 5sin( 4+ln4x)+ C
(j) 3sec(3x4 +3)+ C
(k) 4ln |x3 +2| + C
(l) 4ln |e5x +3| + C =4ln(e5x +3)+ C
2. (a) 8 3
(b) 21 10 (c) 665 6
(d) 1 6 e20 1 6 e5
3. (a) 1 5 1 u2 du
(b) 3 1 u 3 du
Integrals —41