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Mth134 – Calculus II – Volumes and Surfaces of RevolutionWasher method for volume:?V=π(R2-r2)(thickness) R= outer radius r= inner radius thickness = ?x or ?yShell method for volume:?V=2πradiusheight(thickness) radius = distance to axis height = height of piece thickness = ?x or ?ySurface area:?S=2πradius?s radius = distance to axis ?s=1+y'2?x1+x'2?yIn each case, set up the integral to calculate the volume (using either the “washer” or “shell” method, as appropriate) and surface area. All problems will use the same initial area. Only the axis of revolution will change.RegionVolumeSurface AreaRevolve the region about the x-axis. x y y = f (x) x y y = f (x)Revolve the region about the y-axis. x y y = f (x) x y y = f (x)Revolve the region about the line y=3. x y y = f (x) x y y = f (x)Revolve the region about the line y=-1. x y y = f (x) x y y = f (x)Revolve the region about the line x=4. x y y = f (x) x y y = f (x)Revolve the region about the line x=-1. x y y = f (x) x y y = f (x) ................
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