Math 2414 Section 6 4 Volume by Shells Cylindrical Shells Let R be a region bounded by the graph of f the x axis and the lines x a and x b where 0 a b and f x 0 on a b When R is revolved about the y axis a solid is generated whose volume is computed with the slice and sum strategy Volume by the Shell Method f x g x a b Let f and g be continuous functions with on If R is the region bounded y f x y g x by the curves and between the lines x a and x b the volume of the solid generated when R is revolved about the y axis is b V 2p x f x g x dx a An analogous formula for the shell method when R is revolved about the x axis is obtained by reversing the roles of x and y f x sin x 2 Let R be the region bounded by the graph of the x axis and the line x p 2 Find the volume of the solid generated when R is revolved about the y axis Let R be the region in the first quadrant bounded by the graph of y x 2 and the line y 2 First find the volume of the solid generated when R is revolved about the x axis Next find the volume of the solid generated when R is revolved about the line y 2 Restoring Order After working with slices disks washers and shells you may feel somewhat overwhelmed How do you choose a method and which method is best 1 Sketch the graph s of the equation s 2 Determine a slice This will tell you the variable of integration 3 Compare your slice to the axis of revolution a The slice for disks and washers is perpendicular to the axis of revolution b The slice for shells is parallel to the axis of revolution 4 Integrate to find the volume of the solid