Volume by RevolutionLesson Objective - after completing this lesson you will be able to calculate the volume of solids of revolution using disks.
We now consider a specific type of solid known as a solid of revolution. Suppose ƒ is a continuous function with ƒ(x) >=0 on an interval [a,b]. Let R be the region bounded by the graph of ƒ, the x-axis, and the lines x = a and x = b. Now revolve R (out of the page) around the x-axis. As R revolves once about the x-axis, it sweeps out a three-dimensional solid of revolution. The goal is to find the volume of this solid, which may be done using the general slicing method. (For more information, visit Khan Academy) |
Warm-up Example (perpendicular slices): Use the region bounded by
y = x^2, y = -x^2, x = 0 and x = 1 as shown in the graph at right to create a volume by using circular cross sections perpendicular to the x-axis. This is similar to example 2a in the perpendicular slices section. Since we have complete circles we do not need to divide the area by 2. |
Circular cross sections can also be formed by revolving very thin (essentially no width) rectangles about an axis of revolution. These circular cross sections are more commonly called disks. This method can be much easier. In each problem we will use the formula for the area of a circle. We will need to find a formula for the radius of the circular slice.
A volume formed by revolving a region about a line that does not pass through the interior of the region is called a solid of revolution. The line is called the axis of revolution.
If a region is bounded by the axis of revolution, the volume of the solid of revolution is a sum of the volumes of essentially an infinite number of cylindrical disks.
A volume formed by revolving a region about a line that does not pass through the interior of the region is called a solid of revolution. The line is called the axis of revolution.
If a region is bounded by the axis of revolution, the volume of the solid of revolution is a sum of the volumes of essentially an infinite number of cylindrical disks.
Example 3
Set up integrals for the volumes of the solids formed by revolving the region bounded by y = -x^2 + x and y = 0
a) about the x-axis (Discs) b) about y = 1 (Washers) c) about y = 2 (Washers)
Set up integrals for the volumes of the solids formed by revolving the region bounded by y = -x^2 + x and y = 0
a) about the x-axis (Discs) b) about y = 1 (Washers) c) about y = 2 (Washers)
Example 4
Set up integrals for the volumes of the solids formed by revolving the region bounded by y = x^2 and y = x + 2
a) about the x-axis b) about the line y = 4
Set up integrals for the volumes of the solids formed by revolving the region bounded by y = x^2 and y = x + 2
a) about the x-axis b) about the line y = 4
Note: When finding volumes using either discs or washers, you should always sketch your region and draw a representative rectangle. Your figure will help you decide whether to use (the representative rectangles must be perpendicular to the axis of revolution). The figure can also help you choose a method (discs or washers), find limits of integration, and decide which expressions to use for R and r (when using washers). Your integral must be consistent (“all ” with or “all ” with ).
Remember: PerpenDiscular
Remember: PerpenDiscular
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This website was prepared by Brant Breeding in his personal capacity. These materials are not endorsed, approved, sponsored, or provided by or on behalf of the University of Arkansas, Fayetteville.
Images on this page courtesy of Briggs, Gillet, and Schulz and Taylor (full citation found on Reference page).
Please follow this link for our full disclaimer statement.