The disk method can be extended to cover solids of revolution with holes by replacing the representative disk with a representative washer. The washer is formed by revolving a rectangle about an axis, as shown. If r and R are the inner and outer radii of the washer and w is the width of the washer, the volume is given by Volume of washer = π(R 2 – r 2 )w. The Washer Method
To see how this concept can be used to find the volume of a solid of revolution, consider a region bounded by an outer radius R(x) and an inner radius r(x), as shown in Figure 7.19. Figure 7.19 The Washer Method
Washer method If the area between two curves is revolved around an axis, a solid is created that is “hollow” in the center. When slicing this solid, the sections created are washers, not solid disks. The area of the smaller circle must be subtracted from the area of the larger one.
Volume of Washer: Horizontal axis of revolution Slice with rectangle perpendicular to axis Integrate in terms of x Vertical axis of revolution Slice with rectangles perpendicular to axis Integrate in terms of y Outer Radius (R) – top function Inner Radius (r) – bottom function Width – w Right most function Left most function
6) Find the volume of the region bounded by y = x and y = x 2 from x = 0 to x = 1 revolved about the x – axis. 0 Outer radius – R Inner radius – r
7) Find the volume of the region bounded by x = y 2 and x = y 3 from y = 0 to y = 1 revolved about x = 0 (y-axis) 0 1 x = y 2 x = y 3
Are regions always revolved about the x or y axis? Axes of revolutions can be lines as well Radii need adjusted y = 5 R = 5 – x x = -2 R = y + 2 There are two more situations each is similar to these two
8) Find the volume of the region bounded by y = x 2 and y = 4x revolved around the line y = -2 0 4 y = 4x y = x 2 y = -2 Outer Radius f(x) + 2 Inner Radius g(x) + 2