ΔV = (2y)^2 Δx = 4y^2 Δx = 4(9 − x^2 ) Δx.
Now, using symmetry across the y-axis, we find the volume of the solid: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 N7–11
This 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
EXAMPLE 6
Find the volume of a sphere of radius r.
SOLUTION: If the region bounded by a semicircle (with center O and radius r) and its diameter is
revolved about the x-axis, the solid of revolution obtained is a sphere of radius r, as seen in Figure
N7–12.