Definitions●Volume: Letbe a solid that lies betweenand. If the crosssectional area ofSx=ax=bin the plane, throughand perpendicular to the xaxis, is, where A is aSPxx(x)Acontinuous function, then the volume ofisS(x) Δx(x)dxV=limn→∞∑ni=1Ai*=∫baA●Volume of a cylindrical shell:2πrhΔrV=○Can be remembered as[circumference][height][thickness]V=●Shell Method: the volume of the solid, obtained by rotating about the yaxis the regionunder the curvefromto, iswheref(x)y=abπxf(x)dxV=∫ba2b0 ≤a<Examples●Find the volume of the solid obtained by rotating about the yaxis the region bounded byand2xxy=2−30y=○Understand that a typical shell has radius, circumference, and heightxπx2(x)2xxf=2−3○Apply the information to the Shell Method to get the volume■(2πx) (2xx)dxπ(2x)dxV=∫202−3=∫202−x4○Simplify the volume■2π [xx]2π (8)V=214−51520=−532○This yields a final volume ofπV=516●Find the volume of the solid obtained by rotating about the yaxis the region betweenandxy=xy=2○Understand that a typical shell has radius, circumference, and heightxπx2xx−2○Apply the information to the Shell Method to get the volume■(2πx) (x)dx2π(x)dxV=∫10−x2=∫102−x3○Simplify the volumeVolumes by Cylindrical Shells
■(2πy) (1)dy2π(y)dyV=∫10−y2=∫10−y3○Simplify the volume■2π V=2y2−4y410○This yields a final volume ofV=2π■V= 2π [xx3−x44]01○This yields a final volume ofV=π6●Use the cylindrical shells to find the volume of the solid obtained by rotating about thexaxis the region under the curvey= √xfrom[0, 1]○To use shells, relabel the curvey= √xasx=y2○Understand that a typical shell has radiusy, circumference2πy, and height1 −y2○Apply the information to the Shell Method to get the volume
Work & Energy Example 1: Carrying sack of concrete up a ladder while it spills concrete: Weight (Force) 80 pounds at bottom 50 pounds at top 20 feet elevation charge -Write a formula F=80-1.5x x = distance above ground ∑ 𝐹𝐹.∆𝑥𝑥 → 𝑤𝑤=∫𝐹𝐹.𝑑𝑑𝑥𝑥200𝑤𝑤=∫�80−32𝑥𝑥� 𝑑𝑑𝑥𝑥200𝑤𝑤= 80𝑥𝑥 −(32)(𝑥𝑥22)�200𝑤𝑤= 1600−34(400)−0Work=(Force)(Distance) Foot-Pounds Pounds Feet Newton Meters (Nm) Newtons Meters 𝑘𝑘𝑘𝑘.𝑚𝑚2𝑠𝑠2(𝑘𝑘𝑘𝑘)(𝑚𝑚)𝑠𝑠2English Units Metric Units w = 1,300 Foot-Pounds
Example 2: Chain with bucket lowered down a well. Lift it.