3The Washer MethodThe 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 radiiof the washer and w is the width of the washer, the volume is given byVolume of washer = π(R2 – r2)w.
4The Washer MethodTo 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
5Washer methodIf 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.
6Volume of Washer: Horizontal axis of revolution Slice with rectangle perpendicular to axisIntegrate in terms of xVertical axis of revolutionSlice with rectangles perpendicular to axisIntegrate in terms of yWidth – wOuter Radius (R) –top functionInner Radius (r) –bottom functionRight mostfunctionLeft mostfunction
76) Find the volume of the region bounded by y = x and y = x2 from x = 0 to x = 1 revolved about the x – axis.Outer radius – RInner radius – r
87) Find the volume of the region bounded by x = y2 and x = y3 from y = 0 to y = 1 revolved about x = 0 (y-axis)1x = y2x = y3
9Are regions always revolved about the x or y axis? Axes of revolutions can be lines as wellRadii need adjustedx = -2For vertical slices x = numberFor horizontal slices y = numberFormulas stay the sameRadii need adjustedy = 5R = 5 – xR = y + 2There are two more situations each is similar to these two
108) Find the volume of the region bounded by y = x2 and y = 4x revolved around the line y = -2 Outer Radius f(x) + 24y = -2Inner Radius g(x) + 2