Applications of Integration:Volumes by cylindrical sheels(Mainly based on Stewart: Chapter 6,§6.3)Edmund ChiangMATH1014September 9, 20191Volumes1.1Volumes by cylindrical shellsWe consider volumes that are difficult or even impossible to find with the techniques fromthe previous sections. For example, let us consider the following problem.Example .Find the volume of the solid obtained by rotating about the regiony= 2x2-x3andy= 0about they-axis.Figure 1: (Stewart: Volume ofy= 2x2-x3by revolution about they-axis)If we apply the method used in the previous section to find area of vertical thin strip,then giveny, we need to know the corresponding inner and outer radii of the solid. Thatis, we need to solve the cubic equationy= 2x2-x3forx. But the cubic formula is verycomplicated to be really useful. Moreover, if the relation betweenyandxis of fifth degreeor higher, then the celebrated Galois theory from the 19th century already asserted that nosuch formula forxcan be found in general. Thus we look for alternative method instead.