Volumes of Solids of Revolution

You can also use the definite integral to find the volume of a solid that is obtained by revolving a plane region about a horizontal or vertical line that does not pass through the plane. This type of solid will be made up of one of three types of elements—disks, washers, or cylindrical shells—each of which requires a different approach in setting up the definite integral to determine its volume.

Disk method

If the axis of revolution is the boundary of the plane region and the cross sections are taken perpendicular to the axis of revolution, then you use the disk methodto find the volume of the solid. Because the cross section of a disk is a circle with area π r ², the volume of each disk is its area times its thickness. If a disk is perpendicular to the x‐axis, then its radius should be expressed as a function of x. If a disk is perpendicular to the y‐axis, then its radius should be expressed as a function of y.

The volume ( V) of a solid generated by revolving the region bounded by y = f(x) and the x‐axis on the interval [ a, b] about the x‐axis is

  

If the region bounded by x = f(y) and the y‐axis on [ a, b] is revolved about the y‐axis, then its volume ( V) is

  

Note that f(x) and f(y) represent the radii of the disks or the distance between a point on the curve to the axis of revolution.

Example 1: Find the volume of the solid generated by revolving the region bounded by y = x ² and the x‐axis on [−2,3] about the x‐axis.

Because the x‐axis is a boundary of the region, you can use the disk method (see Figure 1).

Figure 1 Diagram for Example 1.

The volume ( V) of the solid is

  

Washer method

If the axis of revolution is not a boundary of the plane region and the cross sections are taken perpendicular to the axis of revolution, you use the washer method to find the volume of the solid. Think of the washer as a “disk with a hole in it” or as a “disk with a disk removed from its center.” If R is the radius of the outer disk and r is the radius of the inner disk, then the area of the washer is π R ² – π r ², and its volume would be its area times its thickness. As noted in the discussion of the disk method, if a washer is perpendicular to the x‐axis, then the inner and outer radii should be expressed as functions of x. If a washer is perpendicular to the y‐axis, then the radii should be expressed as functions of y.

The volume ( V) of a solid generated by revolving the region bounded by y = f(x) and y = g(x) on the interval [ a, b] where f(x) ≥ g(x), about the x‐axis is

  

If the region bounded by x = f(y) and x = g(y) on [ a, b], where f(y) ≥ g(y) is revolved about the y‐axis, then its volume ( V) is

  

Note again that f(x) and g(x) and f(y) and g(y) represent the outer and inner radii of the washers or the distance between a point on each curve to the axis of revolution.

Example 2: Find the volume of the solid generated by revolving the region bounded by y = x ² + 2 and y = x + 4 about the x‐axis.

Because y = x ² + 2 and y = x + 4, you find that

  

The graphs will intersect at (–1,3) and (2,6) with x + 4 ≥ x ² + 2 on [–1,2] (Figure 2).

Figure 2 Diagram for Example 2.

Because the x‐axis is not a boundary of the region, you can use the washer method, and the volume ( V) of the solid is

  

Cylindrical shell method

If the cross sections of the solid are taken parallel to the axis of revolution, then the cylindrical shell method will be used to find the volume of the solid. If the cylindrical shell has radius r and height h,then its volume would be 2π rh times its thickness. Think of the first part of this product, (2π rh), as the area of the rectangle formed by cutting the shell perpendicular to its radius and laying it out flat. If the axis of revolution is vertical, then the radius and height should be expressed in terms of x. If, however, the axis of revolution is horizontal, then the radius and height should be expressed in terms of y.

The volume ( V) of a solid generated by revolving the region bounded by y = f(x) and the x‐axis on the interval [ a,b], where f(x) ≥ 0, about the y‐axis is

  

If the region bounded by x = f(y) and the y‐axis on the interval [ a,b], where f(y) ≥ 0, is revolved about the x‐axis, then its volume ( V) is

  

Note that the x and y in the integrands represent the radii of the cylindrical shells or the distance between the cylindrical shell and the axis of revolution. The f(x) and f(y) factors represent the heights of the cylindrical shells.

Example 3: Find the volume of the solid generated by revolving the region bounded by y = x ² and the x‐axis [1,3] about the y‐axis.

In using the cylindrical shell method, the integral should be expressed in terms of x because the axis of revolution is vertical. The radius of the shell is x, and the height of the shell is f(x) = x ² (Figure 3).

Figure 3 Diagram for Example 3.

The volume ( V) of the solid is