2 x1dxZ20x2x2 2x21 2dxZ20x4 2x2 2dx28 15CheckyourReadingWhyis15thedenominatoroftheresultinexample2TypeIIIntegrals Similarlywede133neatypeIIintegraltobeaniteratedintegraloftheformZdc ID: 110688
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Oftenweevaluatetheinnermostintegralinsidetheintegrandoftheouterinte-gralratherthanwritingtheintegrationsseparately.EXAMPLE2EvaluatethetypeIintegralZ20Zx1x2ydydxSolution:We…rstevaluatetheinnerintegral:Z20Zx1x2ydydx=Z20Zx1x2ydydx=Z20x2y2 2x1dx=Z20x2x2 2�x21 2dx=Z20x4 2�x2 2dx=28 15CheckyourReading:Whyis15thedenominatoroftheresultinexample2?TypeIIIntegrals Similarly,wede…neatypeIIintegraltobeaniteratedintegraloftheformZdcZv(y)u(y)f(x;y)dxdyItisevaluatedbyconsideringytobeconstantintheinnermostintegral,andthenintegratingtheresultwithrespecttoy.EXAMPLE3EvaluatethetypeIIintegralZ10Zyy2(x+y)dxdy2 Letg;hbecontinuouson[a;b]andsuppposethatg(x)h(x)forxin[a;b].IfRisaregioninthexy-planewhichisboundedbythecurvesx=a;x=b;y=g(x),andy=h(x), thenRissaidtobeatypeIregion.Let’s…ndthevolumeofthesolidbetweenthegraphoff(x;y)andthexy-planeoveratypeIregionRwhenf(x;y)0: Todoso,let’snoticethatifthesolidisslicedwithaplaneparalleltothexz-plane,thenitsareaisA(x)=Zh(x)g(x)f(x;y)dy4 Itfollowsthatiffxj;tjg,j=1;:::;n,isataggedpartitionof[a;b];thenthevolumeofthesolidunderthegraphoff(x;y)andovertheregionRisVnXj=1A(tj)xj5 AlimitofsuchsimplefunctionapproximationsyieldsthevolumesbyslicingformulaV=ZbaA(x)dx6 whichisillustratedbelow: Aftercombiningthiswiththede…nitionofA(x);theresultistheiteratedintegralV=Zba"Zh(x)g(x)f(x;y)dy#dx(1)EXAMPLE5Findthevolumeofthesolidunderthegraphoff(x;y)=2�x2�y2overthetypeIregionx=0y=0x=1y=x Solution:Accordingto(1),thevolumeofthesolidisV=Z10Zx0�2�x2�y2dydx7 WeevaluatetheresultingtypeIiteratedintegralby…rstevaluatingtheinnermostintegral:dV=Z102y�x2y�y3 3x0dx=Z102x�4 3x3dx=2 3 CheckyourReading:Whyis2�x2�y2non-negativeovertheregionboundedbyx=0;x=1;y=0;y=x?Explain.VolumesofSolidsoverTypeIIRegions 8 Similarly,ifp(y)q(y)foryin[c;d],thentheregionRinthexy-planeboundedbythecurvesy=c;y=d;x=p(y),andx=q(y), issaidtobeatypeIIregion.Correspondingly,iff(x;y)0forall(x;y)inatypeIIregionR;thenthevolumeofthesolidunderz=f(x;y)andovertheregionRisV=ZdcZq(y)p(y)f(x;y)dxdy(2)EXAMPLE6Findthevolumeofthesolidunderthegraphoff(x;y)=x2+y2overthetypeIIregiony=0x=y2y=1x=ySolution:Todoso,weuse(2)toseethatV=Z10Zyy2�x2+y2dxdyEvaluatingtheinnermostintegralleadstoV=Z10"x3 3+xy2yy2#dy=Z104 3y3�1 3y6�y4dy=3 35Finally,letusnotethatunboundedregionscanleadtoconvergentimproperintegrals.Indeed,unboundedsolidscanhavea…nitevolume.9 EXAMPLE7Findthevolumeofthesolidunderthegraphoff(x;y)=e�x�yoverthe…rstquadrant. Solution:Inthe…rstquadrant,xisin(0;1)andyisin(0;1):Thus,(2)impliesthatV=Z10Z10e�x�ydydxTheinnerintegralisevaluatedasanimproperintegralV=Z10limR!1ZR0e�x�ydydx=Z10limR!1�e�x�0�e�x�Rdx=Z10e�xdxTheresultingintegralisalsoevaluatedasanimproperintegral,lead-ingtoV=limS!1ZS0e�xdx=limS!1�e0�e�R=1Exercises10 IdentifyeachintegralaseithertypeIortypeIIandevaluate:1.R10R10(x+y)dydx2.R20R31x2ydydx3.R20R30xydxdy4.R10R30dydx5.R10Rx0�x2+y2dydx6.R0Rsin(x)0dydx7.R0R0cos(x)dydx8.R0Rx0sin(y)dydx9.R=40Rsec(x)tan(x)0dydx10.R20Rsin(x)0ydydx11.R0Rx0sin(x)dydx12.R10Ry0ex+ydxdy13.R0Rexp(x)0xdydx14.R10Ry0sin�y2dxdy15.R20Ry0ln�y2+1dxdy16.R30R1xeydxdy17.R21Rx20x x2+y2dydx18.R21Rx01 x2+y2dydxSketchtheregionRanddetermineitstype.Then…ndthevolumeofthesolidunderz=f(x;y)andoverthegivenregion.19.f(x;y)=x2+y220.f(x;y)=3R:y=0;y=1R:x=0;x=2x=0;x=1y=0;y=421.f(x;y)=3x+2y22.f(x;y)=6x+yR:x=0;x=1R:x=2;x=3y=0;y=x2y=0;y=ex23.f(x;y)=xy24.f(x;y)=y2R:y=0;y=1R:y=0;y==2x=�y;x=yx=0;x=sin(y)25.f(x;y)=ex+y26.f(x;y)=9�x2�y2R:y=0;y=1R:x=1;x=3x=0;x=1�yy=x;y=x2Thefollowingregionsareunbounded.SketchtheregionRanddetermineitstype.Then…ndthevolumeofthesolidunderz=f(x;y)andoverthegivenregion.27.f(x;y)=1 x2y228.f(x;y)=1 x2+y2R:xin(1;1);yin(1;1)R:x=0;x=229.f(x;y)=x�2e�y30.f(x;y)=1R:xin(1;1)R:xin(0;1)y=0;y=x�2y=x�e�x;y=x+e�x-31.AregularconewithaheighthandabasewithradiusRispositionedsothatitsaxisishorizontal.FindtheareaA(x)ofaverticalcross-sectionof11 FindtheareaA(x)ofacross-sectionatx.Whatisthevolumeofthepyramid?34.TheGreatPyramidis4810tallandhasasquarebasewhichis7560wideoneachside. WhatisthevolumeoftheGreatPyramid?(hint:seeproblem33).35.ExplainwhytheareaofatypeIregioncanbewrittenintheformA=ZbaZh(x)g(x)dydx36.ExplainwhytheareaofatypeIIregioncanbewrittenintheformA=ZdcZq(y)p(y)dxdy37.Explainwhyifa;b;c;anddareallconstant,thenZbaZdcf(x;y)dydx=ZdcZbaf(x;y)dxdywhenbothiteratedintegralsexist.38.Showthatifa;b;c;anddareconstant,thenZbaZdcf(x)g(y)dydx="Zbaf(x)dx#"Zdcg(y)dy#39.UsepropertiesoftheintegraltoshowthatZbaZq(x)p(x)[f(x;y)+g(x;y)]dydx=ZbaZq(x)p(x)f(x;y)dydx+ZbaZq(x)p(x)g(x;y)dydx13 40.UsepropertiesoftheintegraltoshowthatZbaZq(x)p(x)[f(x;y)+g(x;y)]dydx=ZbaZq(x)p(x)f(x;y)dydx+ZbaZq(x)p(x)g(x;y)dydx41.Showthatiffisdi¤erentiableon(a;b),thenforallcin(a;b)wehavef(c)(b�a)+Zbaf(x)dx=ZbaZxcf0(u)dudx42.Showthatiffisdi¤erentiableandiff(0)=0;thenZbaf(x)dx=ZbaZ10f0(ux)dudx14