Many three-dimensional solids can be generated by revolving a curve about the x-axis or y-axis.

First example.

Second example. If we revolve the semi-circle given by

Finding the Volume of a Sphere

Consider a cross-section of the sphere as shown. This cross-section is a circle with radius f(x) and area π[f(x)]². Informally speaking, if we "slice" the sphere vertically into discs, each disc having infinitesimal thickness dx, the volume of each disc is approximately π[f(x)]² dx. If we "add up" the volumes of the discs, we will get the volume of the sphere:

V	=	∫ r -r  π[f(x)]2 dx
	=	∫ r -r  π(r2-x2) dx
	=	π r2x- x3 3 |r | | |-r
	=	π 2 3 r3 -π - 2 3 r3
	=	4 3 πr3,   as expected.

This is called the Method of Discs. In general, suppose y = f(x) is nonnegative and continuous on [a,b]. If the region bounded above by the graph of f, below by the x-axis, and on the sides by x = a and x = b is revolved about the x-axis, the volume V of the generated solid is given by

We can also obtain solids by revolving curves about the y-axis.

Revolving a Region about the y-axis

If we revolve the region enclosed by y = x² and y = 2x, 0 ≤ x ≤ 2, about the y-axis, we generate the three-dimensional solid shown.

Let's find the volume of this solid. If we "slice" the solid horizontally, each slice is a "washer" (it: "rondella"). The outer radius is

Then the volume of the entire solid is given by

∫ 4 0  [π(   __ √ y )2-π(y/2)2] dy	=	∫ 4 0  π y - y2 4  dy
	=	y2 2 - y3 12 | | | | 4 0
	=	π 8- 16 3 -π(0-0)
	=	8π 3 .

This generalization of the Method of Discs is called the Method of Washers ("rondelle": see). As we have seen, these methods may be used when a region is revolved about either axis.

We could have taken a different approach in the previous example:

Look again at the volume of the solid generated by revolving the region enclosed by y = 2x, y = x², 0 ≤ x ≤ 2 about the y-axis. This time, we will view the solid as being composed of a collection of concentric cylindrical shells ("gusci": see; la parola significa anche "conchiglie", ma non in questo caso) of radius x, height 2x-x², and infinitesimal thickness dx. The volume of each shell is approximately given by the lateral surface area ( = 2π·radius·height) multiplied by the thickness:

"Adding up" the volumes of the cylindrical shells,

V	=	∫ 2 0  2πx[2x-x2] dx
	=	∫ 2 0  2π[2x2-x3] dx
	=	4 3 πx3 - 1 2 πx4 | | | | 2 0
	=	32 3 π-8π -(0-0)
	=	8π 3 ,   as found earlier.

This is called the Method of Cylindrical Shells. Suppose f(x), g(x), F(y), G(y) satisfy all the requirements given earlier. Then, for a region revolved about the y-axis,

For a region revolved about the x-axis,

Notes

• In the disc and washer methods, you integrate with respect to the same variable as the axis about which you revolved the region.

• In the method of cylindrical shells, you integrate with respect to the other variable.

Computing volumes using these methods takes some practice. With experience, you will be better able to visualize the solids and determine which method to apply.

Method of Washers:

Method of Cylindrical Shells: