.Questions 25, 26, and 27 in the Practice Exercises illustrate solids with known
cross sections.
When the cross section of a solid is a circle, a typical slice is a disk. When the
cross section is the region between two circles, a typical slice is a washer—a
disk with a hole in it. Both of these solids, which are special cases of solids with
known cross sections, can be generated by revolving a plane area about a fixed
line.
B2. Solids of Revolution
A solid of revolution is obtained when a plane region is revolved about a fixed
line, called the axis of revolution. There are two major methods of obtaining the
volume of a solid of revolution “disks” and “washers.”
DISKS
The region bounded by a curve and the x-axis is revolved around the x-axis,
forming the solid of revolution seen in Figure N7–11. We think of the
“rectangular strip” of the region at the left as generating the solid disk, ΔV (an
element of the volume), shown at the right.
Figure N7–11This disk is a cylinder whose radius, r, is the height of the rectangular strip,
and whose height is the thickness of the strip, Δx. Thus
.