How can we use a definite integral to find the volume of a three-dimensional solid of revolution that results from revolving a two-dimensional region about a particular axis?
In what circumstances do we integrate with respect to instead of integrating with respect to ?
What adjustments do we need to make if we revolve about a line other than the - or -axis?
Just as we can use definite integrals to add the areas of rectangular slices to find the exact area that lies between two curves, we can also use integrals to find the volume of regions whose cross-sections have a particular shape.
In particular, we can determine the volume of solids whose cross-sections are all thin cylinders (or washers) by adding up the volumes of these individual slices. We first consider a familiar shape in Preview Activity 6.2.1: a circular cone.
Consider a circular cone of radius 3 and height 5, which we view horizontally as pictured below. Our goal in this activity is to use a definite integral to determine the volume of the cone.
(b) For the representative slice of thickness that is located horizontally at a location (somewhere between and ), what is the radius of the representative slice? Note that the radius depends on the value of .
Subsection6.2.1The Volume of a Solid of Revolution
A solid of revolution is a three dimensional solid that can be generated by revolving one or more curves around a fixed axis. For example, the circular cone in Preview Activity 6.2.1 is the solid of revolution generated by revolving the portion of the line from to about the -axis. Notice that if we slice a solid of revolution perpendicular to the axis of revolution, the resulting cross-section is a circle.
First, we observe that intersects the -axis at the points and . When we revolve the region about the -axis, we get the three-dimensional solid pictured in Figure 6.2.2.
Figure6.2.2.The solid of revolution in Example 6.2.1.
We slice the solid into vertical slices of thickness between and . A representative slice is a cylinder of height and radius . Hence, the volume of the slice is
slice.
Using a definite integral to sum the volumes of the representative slices, it follows that
.
It is straightforward to evaluate the integral and find that the volume is .
For a solid such as the one in Example 6.2.1, where each slice is a cylindrical disk, we first find the volume of a typical slice (noting particularly how this volume depends on ), and then integrate over the range of -values that bound the solid. Often, we will be content with simply finding the integral that represents the volume; if we desire a numeric value for the integral, we typically use a calculator or computer algebra system to find that value.
If is a nonnegative continuous function on , then the volume of the solid of revolution generated by revolving the curve about the -axis over this interval is given by
First, we must determine where the curves and intersect. Substituting the expression for from the second equation into the first equation, we find that . Rearranging, it follows that
,
and the solutions to this equation are and . The curves therefore cross at and .
When we revolve the region about the -axis, we get the three-dimensional solid pictured at left in Figure 6.2.4.
Figure6.2.4.At left, the solid of revolution in Example 6.2.3. At right, a typical slice with inner radius and outer radius .
Immediately we see a major difference between the solid in this example and the one in Example 6.2.1: here, the three-dimensional solid of revolution isn’t “solid” because it has open space in its center along the axis of revolution. If we slice the solid perpendicular to the axis of revolution, we observe that the resulting slice is not a solid disk, but rather a washer, as pictured at right in Figure 6.2.4. At a given location between and , the small radius of the inner circle is determined by the curve , so . Similarly, the big radius comes from the function , and thus .
To find the volume of a representative slice, we compute the volume of the outer disk and subtract the volume of the inner disk. Since
,
it follows that the volume of a typical slice is
slice.
Using a definite integral to sum the volumes of the respective slices across the integral, we find that
.
Evaluating the integral, we find that the volume of the solid of revolution is .
If and are nonnegative continuous functions on that satisfy for all in , then the volume of the solid of revolution generated by revolving the region between them about the -axis over this interval is given by
In each of the following questions, draw a careful, labeled sketch of the region described, as well as the resulting solid that results from revolving the region about the stated axis. In addition, draw a representative slice and state the volume of that slice, along with a definite integral whose value is the volume of the entire solid. It is not necessary to evaluate the integrals you find.
These two curves intersect when , hence at the point . When we revolve the region about the -axis, we get the three-dimensional solid pictured at left in Figure 6.2.6.
Figure6.2.6.At left, the solid of revolution in Example 6.2.5. At right, a typical slice with inner radius and outer radius .
Note that the slices are cylindrical washers only if taken perpendicular to the -axis. We slice the solid horizontally, starting at and proceeding up to . The thickness of a representative slice is , so we must express the integrand in terms of . The inner radius is determined by the curve , so we solve for and get . In the same way, we solve the curve (which governs the outer radius) for in terms of , and hence . Therefore, the volume of a typical slice is
slice.
We use a definite integral to sum the volumes of all the slices from to . The total volume is
.
It is straightforward to evaluate the integral and find that .
In each of the following questions, draw a careful, labeled sketch of the region described, as well as the resulting solid that results from revolving the region about the stated axis. In addition, draw a representative slice and state the volume of that slice, along with a definite integral whose value is the volume of the entire solid. It is not necessary to evaluate the integrals you find.
Subsection6.2.3Revolving about horizontal and vertical lines other than the coordinate axes
It is possible to revolve a region around any horizontal or vertical line. Doing so adjusts the radii of the cylinders or washers involved by a constant value. A careful, well-labeled plot of the solid of revolution will usually reveal how the different axis of revolution affects the definite integral.
Graphing the region between the two curves in the first quadrant between their points of intersection ( and ) and then revolving the region about the line , we see the solid shown in Figure 6.2.8. Each slice of the solid perpendicular to the axis of revolution is a washer, and the radii of each washer are governed by the curves and . But we also see that there is one added change: the axis of revolution adds a fixed length to each radius. The inner radius of a typical slice, , is given by , while the outer radius is .
Figure6.2.8.The solid of revolution described in Example 6.2.7.
Therefore, the volume of a typical slice is
slice.
Finally, we integrate to find the total volume, and
In each of the following questions, draw a careful, labeled sketch of the region described, as well as the resulting solid that results from revolving the region about the stated axis. In addition, draw a representative slice and state the volume of that slice, along with a definite integral whose value is the volume of the entire solid. It is not necessary to evaluate the integrals you find. For each prompt, use the finite region in the first quadrant bounded by the curves and .
We can use a definite integral to find the volume of a three-dimensional solid of revolution that results from revolving a two-dimensional region about a particular axis by taking slices perpendicular to the axis of revolution which will then be circular disks or washers.
If we revolve about a vertical line and slice perpendicular to that line, then our slices are horizontal and of thickness . This leads us to integrate with respect to , as opposed to with respect to when we slice a solid vertically.
If we revolve about a line other than the - or -axis, we need to carefully account for the shift that occurs in the radius of a typical slice. Normally, this shift involves taking a sum or difference of the function along with the constant connected to the equation for the horizontal or vertical line; a well-labeled diagram is usually the best way to decide the new expression for the radius.
Consider the curve and the portion of its graph that lies in the first quadrant between the -axis and the first positive value of for which . Let denote the region bounded by this portion of , the -axis, and the -axis.
Set up a definite integral whose value is the exact arc length of that lies along the upper boundary of . Use technology appropriately to evaluate the integral you find.
Set up a definite integral whose value is the exact area of . Use technology appropriately to evaluate the integral you find.
Suppose that the region is revolved around the -axis. Set up a definite integral whose value is the exact volume of the solid of revolution that is generated. Use technology appropriately to evaluate the integral you find.
Suppose instead that is revolved around the -axis. If possible, set up an integral expression whose value is the exact volume of the solid of revolution and evaluate the integral using appropriate technology. If not possible, explain why.
Consider the curves given by and . For each of the following problems, you should include a sketch of the region/solid being considered, as well as a labeled representative slice.
Sketch the region bounded by the -axis and the curves and up to the first positive value of at which they intersect. What is the exact intersection point of the curves?
Set up a definite integral whose value is the exact area of .
Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving about the -axis.
Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving about the -axis.
Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving about the line .
Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving about the line .
Determine a definite integral whose value is the area of the region enclosed by the two curves.
Find an expression involving one or more definite integrals whose value is the volume of the solid of revolution generated by revolving the region about the line .
Determine an expression involving one or more definite integrals whose value is the volume of the solid of revolution generated by revolving the region about the -axis.
Find an expression involving one or more definite integrals whose value is the perimeter of the region .