disks.
WASHERS
A washer is a disk with a hole in it. The volume may be regarded as the difference in the volumes of
two concentric disks. As an example, consider the volume of the solid of revolution formed when the
region bounded by the two curves seen in Figure N7–14 is revolved around the x-axis. We think of
the rectangular strip of the region at the left as generating the washer, ΔV (an element of the volume),
shown at the right.
FIGURE N7–14
This washer’s height is the thickness of the rectangular strip, Δx. The washer is a disk whose
outer radius, R, is the distance to the top of the rectangular strip, with the disk of inner radius r (the
distance to the bottom of the strip) removed. Thus:
EXAMPLE 8
Find the volume obtained when the region bounded by y = x^2 and y = 2x is revolved about the x-axis.
SOLUTION: The curves intersect at the origin and at (2, 4), as shown in Figure N7–15. Note that
we distinguish between the two functions by letting (x, y 1 ) be a point on the line and (x, y 2 ) be a
point on the parabola.Washer.