Let V be the volume of the solid S obtained by rotating about the y-axis the region bounded by y= /72x and y3x2. Find V both by slicing and by cylindical shells:(A) The method of cylindrical shells :The circumference of a typical shell =and the height of this shell =The volume V =dx , where a =and b =Therefore V =(B) The method of slicing from Sec(7.2):The volume V = °dy , where a =and b =Thus the volume V =

Consider the provided question, Given function, y=72x and y=3x2 (A) Find the volume of the solid S rotating about y-axis of the region bounded by y=72x and y=3x2 using the cylindrical shell method. First find the intersection point of the given curve, 72x=3x2⇒72x2=3x22⇒72x=9x4⇒9x8-x3=0⇒x=0, x3=8⇒x=0, x=2 Height of the shell = 72x-3x2 Radius of the shell = x Circumference of the shell = 2πr=2πx Now, the volume according to shell method is given by, V=∫ab2πx72x-3x2dxwhere, a=0 and b=2V=∫022πx72x-3x2dx=2π·∫02x72x-3x2dx=2π·∫0272x32-3x3dx=2π72·25·x52-3·x4402=2π72·25·252-3·244-0=2π965-12=72π5Therefore, V=72π5For part (B), Find the volume by method of the slicing form, The sketch of the provided function is given by, The volume using slicing method is, Inner radius of the disc=r1=x1=y272 Since, y=72xOuter radius of the disc=r2=x2=y3 Since, y=3x2Volume=∫abπy32-y2722dywhere, a=0 and b=12Volume=∫012πy32-y2722dy=π·∫012y3-y45184dy=πy22×3-y55×5184012=π1222×3-1255×5184-0=π24-485=725πThus, Volume V=725π