5. The base of a solid is the region betcen the curve y. 2Vsinand the interval [0, w] on the x-axis. The erous-sections perpendicular to the x-axis ana equilalenal triangles with beses running from thes-axis to th4,Thoilie bwppdureExercisesExercisescurvieasshownintheaccompamying figure.and x=1. The cros-sectiones pepedicularto the saxiheAtuecmtheseplanesaresgareswhosediagomalsrunfromthemicircley=V1rtothesemicirclepmV1xVss)2-2)2-bk squres with bases rumning from the x-axis to the curveV=,(=2(1-)=x(n9)-V5an(b) STEP 1) () = (tide) =(2V/in x)(2/in )=4 sln xSTEP 2)a≤0,b=r4ndx=-4=B0+1)2STEP3)VA(x) dx国AEExercises Tho solid les betwem planes perpendietlar to the sratise/3 and x = g/3. The cros-sections perpendicular to thox-axis arcacireulardikx withdiamciersnmning fom thecury tanx to the curvey.sec xIhsquasees uhose bases rin fiom the curvenxtohcurve y = seex3.Solids of Revolution:P(2 x12)(b) STEP 1) A() = (edgn) = (r5xTheDiscMethodSTEP2)a=-55-(2 m2*-1-29)dx=2(V1)4/5STEP 3) V -A(x) ds ()+()(x) dxSTEP.S)V--+(由)(-++(由)(5-号)日EXAMPLE4()RotationAbout thex-axsThe region between the curve y Vx, o ≤ x ≤ 4, and the x-axis irevolved about the x-atis to generate a solid. Find its voltSelutionWe dnw figures shoming the ngion,atypical rndinV3and the getened solid The volume isa[Rn)ParBtetvaThe solid generated by rotuating (orDREandolidof8revohmg)apneregonabouanaxsudutioathlisEEsptanecalledasolidofnerofatioFCURE 6.8:The region (s) ax iolid aoolun(b) inamete4(x) (nadius) [R(x)PNoluebyDiskIXE告人(x)d[R(x)Pd
2016/11/15 6 Exercises 2 2 2 2 ( ) (2 1 ) ( ) 2(1 ) 2 2 diagonal x A x x 1 3 1 2 1 1 8 ( ) 2(1 ) 2 3 3 b a x V A x dx x dx x Exercises Exercises Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley 3. Solids of Revolution: The Disc Method The solid generated by rotating (or revolving) a plane region about an axis in its plane is called a solid of revolution. (1)Rotation About the x-axis
EXAMPLESThecindlePRohEXAMPLEFind the volurcofihesnlenerated by revolving.he tegon bou++y-aAbouttheby y Vr and the lisis y m I,x 4 abouFind ils wolumLine ParallelNE-to the x-axisayio&sTheslenny-V-PaupksSotutisnWe dew figures showing the segion, a typicat radius, and the generated solid(Figure 6.10), The volume i(x) y, w(a?-)m(R(x)P drPsTherefore, the wolu[-[V-1a[[----[[2Vg+1]4thpndindogn国L号-2号+-lrioputEXAMPLE7Find the volume of the solid generated by revolving the region between(3)Rotation Ahouthe y-axsthe y-axis and the curve x = 2/y, I ≤ y ≤ 4, about the yarisTo find the volume ofa solid gemeralod by revolvingaregion berween they-axis andaTcurvex = R(y),e y d, about the y-axis, we use the same method withx replaced byyIn this case, the circular cross-section is(6)4(y) = [mdiusP = [R()P,and the definitionofvolume givesVolume by Disks for Rotatlon About the y-axisSolurtioeWe draw fieurcs sbowine the rcgion. a typisal radius, and the gencratod solidM(y)dy[R(y)Pdy(Figune 6.11). The volume im[R(yIPyRalnkig-Fts1-()0五--4[-4]-3(+jRoutun AlouLbeLine Paralel lo the y-aseEXAMPLE8Find the volume of the solid generated by revolving the region betweentheparabola x y2+ 1 and the line x 3 about the line x =FT-2a/2(ayi)4.SolidsofRevolution:+++aVthe Washer MethodV5We draw figures showing the region, a typical radius, and the generated solidSolution(Figure 6.12). Note that the crosg-sections are perpendicular to the line x = 3 andhave y-coordinates from y V2 to y = V2. The volume ise[RyiPadyy-AV2whear-[2 jfay64eV24+2[44y2 + y'1dyaiealmt5LVSlide 4- 42
2016/11/15 7 (2)Rotation About the Line Parallel to the x-axis (3)Rotation About the y-axis (4)Rotation About the Line Parallel to the y-axis Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 4 - 42 4. Solids of Revolution: the Washer Method