Calculus Section 7.3 Volume by Shells
Homework: page 472 #’s 1, 3, 13,
17, 21, 23, 25, 26
-Find the volume of a solid of revolution using the shell method
-Compare the uses of the disk method and the shell method
The shell method is an alternative method for finding the volume of a solid of revolution. The method is called
the shell method because it uses cylindrical shells to evaluate the volume of a rotation. The shell method does
exactly the same thing as the disk and washer method, but is more powerful and varied in its uses. For
example, to rotate the equation y = x3 + 2x2 – 4x around the y axis would be very difficult because it is not easy
to solve the equation in terms of x. Also, some problems like example 2 on the washer method notes that
required multiple integrals to solve can be written as one.
To derive the shell method, consider a representative rectangle.
Let w be the width of the rectangle, h be the height of the
rectangle, and p be the distance between the axis of revolution
and the center of the rectangle. If you revolve this rectangle
around the axis, you create a cylindrical shell of thickness w.
To find the volume of this shell, think of it as two cylinders.
The radius of the larder cylinder corresponds to the radius of the outer shell, and the radius of the smaller
cylinder to the inner shell. Since p is in the middle of the two, it is the average of the two radii. Thus, the
outer radius = p + w/2 and the inner radius = p – w/2.
The Shell Method
To find the volume of a solid of revolution with the shell method, use one of the following:
Horizontal Axis of Revolution
d
Volume  2  p ( y )h( y )dy
c
Vertical Axis of Revolution
b
Volume  2  p ( x)h( x)dx
a
Notice that the representative rectangles are parallel to the axis of rotation rather than perpendicular. This is
opposite to the disk and washer methods.
Example)
Find the volume of the solid of revolution formed by revolving the region bounded by y = x – x3 and the x-axis
(0 ≤ x ≤ 1) about the y-axis.
Example)
Find the volume of the solid of revolution formed by revolving the region bounded by x  e y and the y-axis
(0 ≤ y ≤ 1) about the x-axis.
2
Example)
Find the volume of the solid formed by revolving the region bounded by the graphs of y = x3 + x + 1, y = 1, and
x = 1 about the line x = 2.