yo6ApPLICATIONSOFDEFINITEINTEGRALSOVERVIEWInChapter 5 we saw that a continuous function over a closed interval has adefinite integral, which is the limitofany Riemann sum for the function.Weproved thatwecould evaluate definite integrals using the Fundamental TheoremofCalculus.Wealsofound that the area under a curve and the area between two curves could be computed asdefinite integrals.In this chapter we extend the applicationsofdefinite integrals to finding volumes,lengthsofplane curves, and areasofsurfacesofrevolution.Wealso use integrals tosolve physical problems involving the work done by a force, the fluid force against aplanar wall, and the locationofan obj ect's centerofmass.6.1VolumesUsingCross-SectionssaCross-sectionS(x)with areaA(x)bxFIGURE6.1A cross-sectionSex)ofthesolid S fonnedbyintersecting S with a planePxperpendicular to the x-axis through thepointxin the interval[a,b].308Inthis section we define volumesofsolids using the areasoftheir cross-sections. A cross-sectionofa solidSis the plane region formed by intersectingSwith a plane (Figure 6.1).Wepresent three different methods for obtaining the cross-sections appropriate to findingthe volumeofa particular solid: the methodofslicing, the disk method, and the washermethod.Suppose we want to find the volumeofa solidSlike the one in Figure 6.1.Webeginby extending the definitionofa cylinder from classical geometry to cylindrical solids witharbitrary bases (Figure 6.2).Ifthe cylindrical solid has a known base areaAand heighth,then the volumeofthe cylindrical solid isVolume=areaXheight=A•h.This equation forms the basis for defining the volumesofmany solids that are not cylin-ders, like the one in Figure6.1.Ifthe cross-sectionofthe solidSat each pointxin the in-terval[a,b]is a regionS(x) ofareaA(x),andAis a continuous functionofx,wecan defineand calculate the volumeofthe solid S as the definite integralofA(x).Wenow show howthis integral is obtained by themethodofslicing.Plane region whosearea we know~=heightCylindrical solid based on regionVolume=base area x height=AhFIGURE6.2The volumeofa cylindrical solid is always defined tobe its base area times its height.

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ySofr'IIIIIIxFIGURE6.3Atypical thin slab in thesolidS.yoPlaneatXk- lApproximatingcylinder basedonS(xk)has height~xk=xk-xk_l/Xk- lPlaneatxk,---The cylinder's baseis the regionS(Xk)with areaA(xk)l~XNUfTOSCALEFIGURE6.4The solid thin slab inFigure 6.3 is shown enlarged here.Itisapproximated by the cylindrical solid withbaseS(Xk)having areaA(Xk)and heightIlxk=Xk-Xk- l ·6.1VolumesUsingCross-Sections309SlicingbyParallel PlanesWepartition[a,b]into subintervalsofwidth (length)fllkand slice the solid, as wewould a loafofbread, by planes perpendicular to the x-axis at the partition pointsa=Xo<Xl< ... <Xn=b.The planesPXk'perpendicular to the x-axis at the parti-tion points, sliceS

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