Disk MethodLet’s rotate the region under f(x) from a to b about x-axis.Separate the area into rectangles of equal length.Revolve each separately to form many disk.Vof 1 disk= π(radius)2•(thickness)Total volume = sum of volumes of infinitely thin individual disks.
Example 1Find the volume of the solid generated when the region bounded by y = 3x - x2and the x-axis is revolved around the x-axis.1. Graph the region and identify the thickness (dx or dy)2. Express the radius for one disk in terms of x (if dx) or y (if dy).3. Integrate: Use Disk MethodSince the region is revolved around x-axis → Thickness of a disk = dx.81π/10
Example 2The region bounded by the y-axis and the graphs of y = x3/8, y = 0, and y = 8 is revolved about the y-axis. Find the volume of the resulting solid.Thickness of disks: dyRadius of disks:Why Because ?
Example 3Sketch the region R bounded by the graph of y = sin(2x) and y = 0 from 0 to π, and find the volume of the solid generated if R is revolved around the x-axis.As the graph of y = sin(2x) is symmetric with respect to the point (pi/2,0), the area of R is2* Integral (x from 0 to pi/2)sin2(x) = (1-cos(2x))/2