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Name: _________________ Period: ______ Date: ________________ AP Calc AB

Mr. Mellina

Chapter 8: Applications of Definite Integrals

Sections:

v 8.1 Integral as Net Change v 8.2 Areas in the Plane

v 8.3 Volumes

HWSets

SetA(Section8.1)Pages390&391,#’s1,3,7,11-16,19,21

SetB(Section8.1)Page390&392,#’s2&6(nocalc),31-36(calc)

SetC(Section8.2)Pages399-401,#’s1&2(calc),3-6(nocalc),52,53,55(nocalc),and54(calc)

SetD(Section8.3)Page400,#’s9,10,13,15,36,38(nocalc)

SetE(Section8.4)Page437,#’s53-55,Page412#39

SetF(Section8.4)Page411#’s7-15(DiskandWasherMethod)

SetG(Section8.4)Page411#’s27,29,31(DiskandWasherMethod)

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8.1 Integrals as Net Change Topics

v Linear Motion Revisited v General Strategy

v Consumption over time v Net change from data

Warm Up! Findallvaluesofx(ifany)atwhichthefunctionchangessignonthegiveninterval.Sketchanumberlinegraphoftheinterval,andindicatethesignofthefunctiononeachsubinterval.a. sin 2𝑥on −3, 2 b. 𝑥) − 3𝑥 + 2on −2, 4 Example1Oilisleakingfromastoragetankatarateof𝑟 𝑡 = 4000𝑒1.34liters/day.a. Howmuchoilwillleakinthefirst5days? b. Howmuchoilwillleakinthenext5days?c. Howmuchoilwillleakinthefirst50days?

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Example2

Velocityisnottheonlyrateinwhichyoucanintegratetogetatotal.Wecanalsouseintegralstorepresent“consumption”overacertaintimeperiod.Infact,ifyouweregivenafunctionthatgavethenumberofticketsperhourthatthepolicewroteeachday,andyouwantedtofindthetotalnumberofticketsina24-hourperiod,youcouldintegrate.

a. Usingtheabovescenario,supposetherateofticketwritingisgivenby𝑁(ℎ).Writeanintegraltofindthetotalnumberofticketswrittenin2days.

To find ___________________, find ∫ 𝑣(𝑡);< 𝑑𝑡.

To find a new location or position, realize that _________________ = _________________ + _________________ _________________ = _________________ + _________________ To find total distance, find >∫ 𝑣(𝑡);

< > 𝑑𝑡 or to find when the object is moving in the negative direction, break the integral into pieces and subtract the value of the intergral for the area under the curve.

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Example3

ThetideremovessandfromSandyPointBeachataratemodeledbythefunctionRgivenbelow.ApumpingstationaddssandtothebeachataratemodeledbythefunctionSgivenbelow.Both𝑅(𝑡)and𝑆(𝑡)haveunitsofcubicyardsperhourandtismeasuredinhoursfor0 ≤ 𝑡 ≤ 6.Attime𝑡 = 0,thebeachcontains2,500cubicyardsofsand.

𝑅 𝑡 = 2 + 5 sin DE4)F

𝑆 𝑡 = GF4GH34

a. Howmuchsandwillthetideremovefromthebeachduringthis6-hourperiod?Indicateunitsofmeasure.

b. Writeanexpressionfor𝑌(𝑡),thetotalnumberofcubicyardsofsandonthebeachattimet.

c. Findtherateatwhichthetotalamountofsandonthebeachischangingattimet=4.

d. For0 ≤ 𝑡 ≤ 6,atwhattimetistheamountofsandonthebeachaminimum?Whatistheminimumvalue?Justifyyourresponse.

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Example4

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8.1APQuestions(CalculatorAllowedonall)2003AB2FormB2002AB2FormB2004AB2FormB

2010AB12006AB22005AB2FormB

2004AB1

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8.2 Areas in the Plane Topics

v Areas between Curves v Area Enclosed by Intersecting Curves v Boundaries with changing functions

Warm Up! Findtheareabetweenthex-axisandthegraphofthegivenfunctionoverthegivenintervalwithoutacalculator.a. 𝑦 = sin 𝑥over 0, 𝜋 Findthex-andy-coordinatesofallpointswherethegraphsofthegivenfunctionsintersectwithoutacalculator.b. 𝑦 = )L

LMHGand𝑦 = 𝑥3

Recall Riemann Sums The area under a curve can be approximated through the use of Riemann sums: We can break the area into rectangles. Consider the one rectangle drawn. It’s height is given by the function value of the curve at the right endpoint and the width is given as ∆𝑥. The area under the curve then is approximately the sum of the areas of ALL the rectangles just like this one. Area ≈

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Example1Drawarectangularstrip.Whatistheheightandwidthofyourrectangle?Wouldtheheightandwidthoftherectanglestripbedifferentifyoudrewitinadifferentplace?

Recall Riemann Sums As the number of rectangles, n, increases, the approximated area gets closer to the actual area, so we say Area under the curve =

limR→T

U 𝑓(𝑐𝑘)𝑛

𝑘=1∆𝑥[ = \ 𝑓(𝑥)𝑑𝑥

;

<

We can apply this same concept to the area between curves. Consider the two functions f and g below.

The area between the curves is approximately the _____ of all these rectangles. We can write this as How can we get closer to the ACTUAL area between the curves? If we let the number of rectangles approach infinity, then we have

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Example2:AreaBetweenTwoCurvesFindtheareaoftheregionbetweenthetwocurves.a. 𝑓 𝑥 = 2 − 𝑥)and𝑔 𝑥 = 𝑥b. 𝑦 = 𝑥)and𝑦 = 𝑒Lontheinterval[0,3].

Area of a Region Between Two Curves If f and g are continuous on [a, b] and 𝑔(𝑥) ≤ 𝑓(𝑥) for all x in [a, b], then the area of the region bounded by the graphs of f and g and the vertical lines _______ and ______ is A =

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c. 𝑓 𝑥 = 3𝑥3 − 𝑥) − 10𝑥and𝑔 𝑥 = −𝑥) + 2𝑥d. InQuadrant1:𝑦 = 𝑥,𝑦 = 𝑥 − 2,andthex-axis.

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e. 𝑦 = 𝑥) + 2,𝑦 = −𝑥,𝑥 = 0,and𝑥 = 1f. 𝑥 = 3 − 𝑦)and𝑥 = 𝑦 + 1

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8.3 Volumes Topics

v Volume as an Integral v Square Cross Sections v Circular Cross Sections

v Cylindrical Shells v Other Cross Sections

Warm Up! Giveaformulafortheareaoftheplaneregionintermsofthesinglevariablex.a. Asquarewithsidesoflengthx b. Asquarewithdiagonalsoflengthxc. Asemicircleofradiusx d. Asemicircleofdiameterxe. Anequilateraltrianglewithsidesof f. Anisoscelesrighttrianglewithlegsof

lengthx lengthx

Just like in the last section where we found the area of one arbitrary rectangular strip and used an integral to add up the areas of an infinite number of infinitely thin rectangles, we are going to apply the same concept to finding volume. The key… find the volume of ONE arbitrary ___________ and use an integral to add up the volumes of an infinite number of infinitely thin slices.

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Example1a. Findthevolumeofthefollowingsquare“slice”.Sincemostofthe“slices”wewillbe

dealingwithwillhaveathicknessofdx,wewillusethatsamethicknesshere. b. Findthevolumeofthefollowingsemicircular“slice”

Day 1: Volumes of Solids with Known Cross Sections (aka ) First question… what is a cross section? Imagine a loaf of bread. Now imagine the shape of a slice through the loaf of bread. This shape would be a cross section. Technically a cross section of a three dimensional figure is the intersection of a ___________ and that __________. It would be like cutting an object and looking at the face of where you just cut. The cross sections we will be dealing with are almost entirely ____________________ to the x-axis. Here’s the basic idea… you will be given a region defined by a number of functions. We will graph that region on an x and y-axis. Then we will lay the region ______ and build upon that region which has the ______ cross section no matter where you slice it. Second question… How do we find the volume of this lice that has been created to have a similarly shaped cross section, even though each cross section may have a different size? We get to use Calculus, of course! But first, we need to know how to find the __________ of a prism. Even though every shape may be different, we can find the volume of a prism by finding the area of the base times the ____________. The “height” of our prisms will be the thickness of the slices. Once you know the volume of one slice, you just use an ____________ to add the volumes of all the slices to get the volume of the solid.

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Example2Thebaseofasolidistheregioninthefirstquadrantenclosedbytheparabola𝑦 = 4𝑥),theline𝑥 = 1,andthex-axis.Eachplanesectionofthesolidperpendiculartothex-axisisasquare.Thevolumeofthesolidisa. DE

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b. G^E

F

c. D

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d. G^

F

e. ^D

F

Example3Thebaseofasolidistheregioninthefirstquadrantboundedbythex-axis,they-axis,andtheline𝑥 + 2𝑦 = 8.Ifthecrosssectionsofthesolidperpendiculartothex-axisaresemicircles,whatisthevolumeofthesolid?a. 12.566b. 14.661c. 16.755d. 67.021e. 134.041

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Example4Thebaseofasolidistheregioninthefirstquadrantenclosedbythegraphof𝑦 = 2 − 𝑥)andthecoordinateaxes.Ifeverycrosssectionofthesolidperpendiculartothey-axisisasquare,thevolumeofthesolidisgivenbya. 𝜋 2 − 𝑦 )𝑑𝑦)

` b. 2 − 𝑦 𝑑𝑦)

`

c. 𝜋 2 − 𝑥) )𝑑𝑥)`

d. 2 − 𝑥) )𝑑𝑥)`

e. 2 − 𝑥) 𝑑𝑥)`

Example5WhatisthevolumeofthecylindershowniftheheightoftherectangleisconsideredRandthewidthoftherectangleisdx?

Day 2: Volumes of Solids of Revolution: The Disc Method In finding the area of a region we drew an arbitrary representative rectangle. Keep with the same idea, if we revolve a rectangle around a line, it forms a ___________, as shown below. Keys to Disc Method

1. Always rotate around an ________ of the region; it must touch the region. 2. “Slice” or rectangle is _____ to axis or revolution creating a cylinder.

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Example6Drawanappropriaterectangularstripandfindthevolumeofthesolidformedbyrevolvingtheregionaboutthex-axis.(www.shodor.org/interactive/activites/functionrevolution/)

Just like we did in finding the area, as we increase the number of rectangles to infinity, the width of each rectangle becomes infinitely small and we denote this dx (if it is a vertical strip) or dy (if it is a horizontal strip). We then use an integral to sum the volume of every one of these infinitely thin cylinders. The Disc Method To find the volume of a solid of revolution with the disc method, use one of the following: Horizontal Axis of Revolution Vertical Axis of Revolution V = V = Where R(x) and R(y) are the ____________ of your representative rectangular strips

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Example7Findthevolumeofthesolidformedbyrevolvingtheregionaboutthey-axis.(Drawarepresentativerectangularstrip)Example8Findthevolumeofthesolidgeneratedbyrevolvingtheregionboundedbythegraphsoftheequations𝑥𝑦 = 6, 𝑦 = 2, 𝑦 = 6and𝑥 = 6abouttheindicatedlines.Sketchtheregionformed,anddrawarepresentativerectangularstripforeachsolid.a. thelinex=6 b. theliney=6

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Example9a. Sketchthefigureformedbyrotatingtherectanglearoundthegivenline.Doyousee

whyit’scalledthewashermethod?Jb. Whatisthevolumeofthefigureformedabove?

Day 3: Volumes of Solids of Revolution: The Washer Method For the disc method, the area we revolved had to be connected to the area of rotation and the representative rectangle had to be perpendicular to the axis of revolution. Keys to the Washer Method

1. Rectangle ____ to the axis of revolution. 2. Rectangle not always connected to _______.

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Example10Setupanintegral,butdonotsolvetofindthevolumeofthesolidgeneratedbyrevolvingtheregionboundedbythegraphsoftheequationsabouttheindicatedlines.

𝑦 = 2𝑥); 𝑦 = 0; 𝑥 = 2a. They-axis b. Thex-axisc. Theliney=8 d. Thelinex=2

We will call the outer radius ___, and we will call the inner radius ___. The height of the cylinder formed is just the ___________ of the strip. Just like before, if we have an infinitely thin strip, this distance will be denoted dx (if it is a vertical strip) and dy (if it is a horizontal strip). The volume of the solid formed by revolving a region around the axis using the Washer Method is given by V =

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8.3APQuestions2001AB1(CalculatorAllowed)2002AB1(CalculatorAllowed)2007AB1(CalculatorAllowed)2012AB2(CalculatorAllowed)2013AB5(NoCalculator)