Does the chapter title leave you in a cold sweat? Don’t worry. You’re not alone. This chapter covers a
topic widely seen as one of the most difficult on the AP Exam. There is always a volume question on the
test. The good news is that you’re almost never asked to evaluate the integral—you usually only have to
set it up. The difficulty with this chapter, as with Chapter 16, is that there aren’t any simple rules to
follow. You have to draw the picture and figure it out.
In this chapter, we’re going to take the region between two curves, rotate it around a line (usually the x- or
y-axis), and find the volume of the region. There are two methods of doing this: the washers method and
the cylindrical shells method. Sometimes you’ll hear the washers method called the disk method, but a
disk is only a washer without a hole in the middle.
WASHERS AND DISKS
Let’s look at the region between the curve y = and the x-axis (the curve y = 0), from x = 0 to x = 1,
and revolve it about the x-axis. The picture looks like the following:
If you slice the resulting solid perpendicular to the x-axis, each cross-section of the solid is a circle, or
disk (hence the phrase “disk method”). The radii of the disks vary from one value of x to the next, but you
can find each radius by plugging it into the equation for y: Each radius is . Therefore, the area of each
disk is