Calculus Section 7.2 Volume by Disk Method
Homework: page 463 #’s 13, 8-10, 11c, 12cd
-Find the volume of a solid of revolution using the disk method
We have defined the integral of a function as a way to find the ______________________. However, there
are other applications of the integral. One application that is related to area is finding the volume of a 3dimensional surface.
If a region in the plane is revolved about a line, the resulting solid is called a solid of revolution, and the line
w
that it revolved around is called the axis of revolution.
w
For example, if you take a simple rectangle of height, R,
and width, w, you can create a disk through rotation.
The volume of this disk can be found by using:
Axis of revolution
Volume of disk = (area of disk)(width of disk)
= (πR2)(w)
where R is the radius of the disk and w is the width.
R
R
We can extend this single disk to find the volume of a general solid of revolution. Consider a solid of
revolution formed by revolving a plane region around an axis. The resulting figure is 3-dimensional and can be
approximated using n disks of varying size.
Logically, this approximation becomes better as the limit of the width of each disk approaches 0. This means
that we could represent the volume of the shape by finding the summation of all the disks; which means that
an integral can do the same thing.
Known formula
Single Representative Disk
New Integration Formula
The Disk Method
To find the volume of a solid of revolution with the disk method, use one of the following formulas:
Horizontal Axis of Revolution
Vertical Axis of Revolution
Example) Using the Disk Method x-axis
Find the volume of the solid formed by revolving the region bounded by the graph of f ( x)  sin x and the
x-axis from [0, π] about the x-axis.
Example) Using the Disk Method y-axis
Find the volume of the solid formed by revolving the region bounded by the graph of y = x2 + 1 and the y-axis
for 1 ≤ y ≤ 5 about the y-axis.
Example) Revolving About a Line That is Not a Coordinate Axis
Find the volume of the solid formed by revolving the region bounded by f ( x)  2  x 2 and g(x) = 1 about the
line y = 1.