To create the cone, consider revolving a line through the origin about the
-axis.

The equation of this line is .

We will integrate over the interval

The definite integral that gives the volume of the cone is:

The volume of the cone is: .

We use disks, washers and shells to find the volume of a solid of revolution.

A special type of solid whose cross-sections are familiar geometric shapes is the solid of revolution. We will use the idea of the last section, namely that volume can be found by integrating cross-sectional area, to find the volume of such a solid.

Solid of revolution A **solid of revolution** is a solid obtained by revolving a region in
the -plane about a line called the **axis of revolution**.

Cylinders, cones and spheres are examples of well known solids which are solids of revolution. For simplicity, we will only consider horizontal and vertical lines for our axis of revolution. Furthermore, we will assume that our region in the plane is bounded by functions of . In the event that we have a region bounded by functions of , we can interchange the variables and throughout the problem to convert the problem to the setting of functions of .

The main formulas we will need from geometry are the area of a disk, area of a washer and the lateral surface area of a cylinder:

To see how these particular shapes come into play, imagine revolving a vertical segment about the axis of revolution. If the axis of revolution is horizontal and the vertical segment touches the axis at one endpoint, this will produce a disk (see the figure below).

If the vertical segment does not touch the horizontal axis of revolution, then revolving it will create a washer.

Finally, if the axis of revolution is vertical, then revolving the vertical segment about the axis of revolution will produce the lateral surface of a cylinder (see the figure below).

example 1 Find the volume of a right, circular cone with height 4, and base radius
2.

This cone can be obtained by revolving the the region between the line and the -axis from to about the -axis.

To begin, we sketch the region (see below) and we draw a vertical line segment in the region at . Revolving this segment about the axis of revolution (the -axis) creates a disk. The area of this disk is where the radius, , is given by We find the volume by integrating area: Computing this integral gives

This cone can be obtained by revolving the the region between the line and the -axis from to about the -axis.

To begin, we sketch the region (see below) and we draw a vertical line segment in the region at . Revolving this segment about the axis of revolution (the -axis) creates a disk. The area of this disk is where the radius, , is given by We find the volume by integrating area: Computing this integral gives

(problem 1a) Find the formula for the volume of a right, circular cone with height ,
and base radius . Include a rough sketch of the region being revolved.

To create the cone, consider revolving a line through the origin about the
-axis.

The equation of this line is .

We will integrate over the interval

The definite integral that gives the volume of the cone is:

The volume of the cone is: .

(problem 1b) Find the formula for the volume of a right, circular cone with height ,
and base radius . Include a rough sketch of the region being revolved.

To create the cone, consider revolving a line through the origin about the
-axis.

The equation of this line is .

We will integrate over the interval

The definite integral that gives the volume of the cone is:

The formula for the volume of a cone is: .

(problem 1c) Find the formula for the volume of the frustum of the right, circular
cone obtained by revolving the region bounded by the lines and about the -axis.
Include a rough sketch of the region being revolved.

The radius of the disk at is

We will integrate over the interval

The definite integral that gives the volume of the solid is:

The volume of the solid is .

example 2 Find the volume of a sphere of radius 2.

We can think of this sphere as the solid of revolution obtained by revolving the region between the semi-circle and the -axis about the -axis. To begin, we sketch the region including a vertical line segment in the region at (see below). Revolving this segment about the axis of revolution (the -axis) creates a disk. The area of this disk is where the radius, , is given by We find the volume by integrating area: Computing this integral gives

We can think of this sphere as the solid of revolution obtained by revolving the region between the semi-circle and the -axis about the -axis. To begin, we sketch the region including a vertical line segment in the region at (see below). Revolving this segment about the axis of revolution (the -axis) creates a disk. The area of this disk is where the radius, , is given by We find the volume by integrating area: Computing this integral gives

(problem 2a) Find the formula for the volume of a sphere of radius . Include a rough
sketch of the region being revolved.

To create the sphere, consider revolving a semi-circular region centered at the origin
about the -axis.

The equation of this semi-circle is .

We will integrate over the interval

The definite integral that gives the volume of the sphere is:

The volume of the sphere is: .

(problem 2b) Find the formula for the volume of a sphere of radius . Include a rough
sketch of the region being revolved.

To create the sphere, consider revolving a semi-circular region centered at the origin
about the -axis.

The equation of this semi-circle is .

We will integrate over the interval

The definite integral that gives the volume of the sphere is:

The formula for the volume of a sphere with radius is: .

example 3 Find the volume of the solid obtained by revolving the region bounded by
the graphs of and about the line .

We first find the points of intersection of the parabola and the horizontal line : Next, we sketch the region including a vertical line segment in the region at (see below).

We first find the points of intersection of the parabola and the horizontal line : Next, we sketch the region including a vertical line segment in the region at (see below).

Revolving this segment about the line creates a disk (which is a cross-section of the solid). The area of this disk is where the radius is given by

We find the volume by integrating the area of a cross-section: Computing this integral gives

(problem 3a) Find the volume of the solid obtained by revolving the region bounded
by the graphs of and about the line . Include a rough sketch of the region being
revolved.

The radius of the disk at is

We will integrate over the interval

The definite integral that gives the volume of the solid is:

The volume of the solid is .

(problem 3b) Find the volume of the solid obtained by revolving the region bounded
by the graphs of and about the line . Include a rough sketch of the region being
revolved.

The radius of the disk at is

We will integrate over the interval

The definite integral that gives the volume of the solid is:

The volume of the solid is .

In the next series of examples, the region will not border the axis of revolution. In this situation, the resulting cross-sections are washers rather than disks. The area of a washer with inner radius, , and outer radius, , is given by

We now use this to calculate volumes of solids of revolutions about the -axis in which the region does not border the axis.

example 4 Find the volume of the solid obtained by revolving the region bounded by
the graphs of , and about the -axis.

First, note that the curves and intersect at .

First, note that the curves and intersect at .

Thus, to compute the volume of the resulting solid, fix a value of between and and consider a vertical segment in the region at this -value.

Revolving this individual segment about the -axis yields a washer with inner radius, , outer radius, .

The area of this washer is then The volume of the solid can be obtained by integrating the area of a cross-section as ranges from to :

(problem 4) Find the volume of the solid obtained by revolving the region between
and from to about the -axis. Include a rough sketch of the region being
revolved.

The outer radius of the washer at is

The inner radius of the washer at is

We will integrate over the interval

The definite integral that gives the volume of the solid is:

The volume of the solid is .

example 5 Find the volume of the solid obtained by revolving the region bounded by
the graphs of , and about the -axis.

We begin with a sketch of the region.

We begin with a sketch of the region.

To compute the volume of the resulting solid, fix a value of between 0 and and consider a vertical segment in the region at this -value.

Revolving this individual segment about the -axis yields a washer with inner radius, and outer radius, .

The area of this washer is then The volume of the solid can be obtained by integrating the area of a cross-section as ranges from 0 to :

(problem 5) Find the volume of the solid obtained by revolving the region between
and from to about the -axis. Include a rough sketch of the region being
revolved.

The outer radius of the washer at is

The inner radius of the washer at is

We will integrate over the interval

The definite integral that gives the volume of the solid is:

The volume of the solid is .

example 6 Find the volume of the solid obtained by revolving the region bounded by
the graphs of and about the line .

The -coordinates of the points of intersection of the curves are: Next we make a sketch of the region including a vertical segment at (see below).

The -coordinates of the points of intersection of the curves are: Next we make a sketch of the region including a vertical segment at (see below).

Revolving this vertical segment about the axis of revolution makes a washer.

The area of this cross-sectional washer is where Squaring these radii gives and The volume of the solid is found by integrating the area of this cross-sectional washer:

Computing this integral gives,

(problem 6) Find the volume of the solid obtained by revolving the region between
and from to about the line . Include a rough sketch of the region being
revolved.

The outer radius of the washer at is

The inner radius of the washer at is

We will integrate over the interval

The definite integral that gives the volume of the solid is:

The volume of the solid is .

We now find the volume of solids of revolution obtained by revolving a region about a vertical axis.

example 7 Find the volume of the solid obtained by revolving the region bounded by
the graphs of , and about the -axis.

We begin with a sketch of the region.

We begin with a sketch of the region.

To compute the volume of the resulting solid, fix a value of between 0 and 1 and consider a vertical segment in the region at this -value.

Revolving this individual segment about the -axis yields a cylindrical shell with radius, height, , and thickness, .

The surface area of this shell is: The volume of the solid can be obtained by integrating the area of the shells as ranges from 0 to 1:

(problem 7) Find the volume of the solid obtained by revolving the region
between and about the -axis. Include a rough sketch of the region being
revolved.

The radius of the cylindrical shell at is

The height of the cylindrical shell at is

We will integrate over the interval

The definite integral that gives the volume of the solid is:

The volume of the solid is .

example 8 Find the volume of the solid obtained by revolving the region bounded by
the graphs of , and about the -axis.

We begin with a sketch of the region.

We begin with a sketch of the region.

To compute the volume of the resulting solid, fix a value of between 0 and 1 and consider a vertical segment in the region at this -value.

Revolving this individual segment about the -axis yields a cylindrical shell with radius, and height, .

The surface area of the shell is: The volume of the solid can be obtained by integrating the area of the shells as ranges from 0 to 1:

(problem 8) Find the volume of the solid obtained by revolving the region
between and about the -axis. Include a rough sketch of the region being
revolved.

The radius of the cylindrical shell at is

The height of the cylindrical shell at is

We will integrate over the interval

The definite integral that gives the volume of the solid is:

The volume of the solid is .

example 9 Find the volume of the solid obtained by revolving the region bounded by
the graphs of and about the line .

We begin with a sketch of the region.

We begin with a sketch of the region.

Note that the parabola crosses the -axis at and . To compute the volume of the resulting solid, fix a value of between 0 and 1 and consider a vertical segment in the region at this -value.

Revolving this individual segment about the line yields a cylindrical shell with radius, and height, .

The surface area of the shell is: The volume of the solid can be obtained by integrating the area of the shells as ranges from 0 to 1:

(problem 9) Find the volume of the solid obtained by revolving the region
between and about the line . Include a rough sketch of the region being
revolved.

The radius of the cylindrical shell at is

The height of the cylindrical shell at is

We will integrate over the interval

The definite integral that gives the volume of the solid is:

The volume of the solid is .

Here is a video on the method of washers and the method of cylindrical shells

_

(problem 10) Find the volume of the solid obtained by revolving the region bounded
by the graphs of and about the line . Include a rough sketch of the region being
revolved.

The method we will use is:

The method we will use is:

Disks Washers Shells

The formula associated with this method is:

The left and right endpoints of integration are:

and .

The radius and height as functions of are:

and .

The definite integral that represents the volume is given by:

The volume of the solid is .

(problem 11) Find the volume of the solid obtained by revolving the region bounded
by the graphs of and about the line . Include a rough sketch of the region being
revolved.

The method we will use is:

The method we will use is:

Disks Washers Shells

The formula associated with this method is:

The left and right endpoints of integration are:

and .

The outer radius, , and inner radius, , as functions of are:

and .

The definite integral that represents the volume is given by:

The volume of the solid is .

(problem 12) Find the volume of the solid obtained by revolving the region bounded
by the graphs of , and about the -axis. Include a rough sketch of the region being
revolved.

The method we will use is:

The method we will use is:

Disks Washers Shells

The formula associated with this method is:

The left and right endpoints of integration are:

and .

The radius, , as a function of is:

The definite integral that represents the volume is given by:

The volume of the solid is .

(problem 13) Find the volume of the solid obtained by revolving the region bounded
by the graphs of , and about the -axis. Include a rough sketch of the region being
revolved.

The method we will use is:

The method we will use is:

Disks Washers Shells

The formula associated with this method is:

The left and right endpoints of integration are:

and .

The radius, , as a function of is:

The definite integral that represents the volume is given by:

The volume of the solid is .

(problem 14) Find the volume of the solid obtained by revolving the region bounded
by the graphs of and about the line . Include a rough sketch of the region being
revolved.

The method we will use is:

The method we will use is:

Disks Washers Shells

The formula associated with this method is:

The left and right endpoints of integration are:

and .

The outer radius, , and inner radius, , as functions of are:

and .

The definite integral that represents the volume is given by:

The volume of the solid is .

(problem 15) Find the volume of the solid obtained by revolving the region bounded
by the graphs of , and about the -axis. Include a rough sketch of the region being
revolved.

The method we will use is:

The method we will use is:

Disks Washers Shells

The formula associated with this method is:

The left and right endpoints of integration are:

and .

The radius, , as a function of is:

The definite integral that represents the volume is given by:

The volume of the solid is .

(problem 16) Find the volume of the solid obtained by revolving the region bounded
by the graphs of and about the -axis. Include a rough sketch of the region being
revolved.

The method we will use is:

The method we will use is:

Disks Washers Shells

The formula associated with this method is:

The left and right endpoints of integration are:

and .

The radius and height as functions of are:

and .

The definite integral that represents the volume is given by:

The volume of the solid is .

(problem 17) Find the volume of the solid obtained by revolving the region bounded
by the graphs of and about the line . Include a rough sketch of the region being
revolved.

The method we will use is:

The method we will use is:

Disks Washers Shells

The formula associated with this method is:

The left and right endpoints of integration are:

and .

The outer radius, , and inner radius, , as functions of are:

and .

The definite integral that represents the volume is given by:

The volume of the solid is .

(problem 18) Find the volume of the solid obtained by revolving the region bounded
by the graphs of and about the line . Include a rough sketch of the region being
revolved.

The method we will use is:

The method we will use is:

Disks Washers Shells

The formula associated with this method is:

The left and right endpoints of integration are:

and .

The radius and height as functions of are:

and .

The definite integral that represents the volume is given by:

The volume of the solid is .