Solids of Revolution

If a region in a plane is revolved around a line in that plane, the resulting solid is called a solid of revolution, as shown in the following figure.

This figure has three graphs. The first graph, labeled “a” is a region in the x y plane. The region is created by a curve above the x-axis and the x-axis. The second graph, labeled “b” is the same region as in “a”, but it shows the region beginning to rotate around the x-axis. The third graph, labeled “c” is the solid formed by rotating the region from “a” completely around the x-axis, forming a solid.

Figure 6.15 (a) This is the region that is revolved around the x-axis. (b) As the region begins to revolve around the axis, it sweeps out a solid of revolution. (c) This is the solid that results when the revolution is complete.

Solids of revolution are common in mechanical applications, such as machine parts produced by a lathe. We spend the rest of this section looking at solids of this type. The next example uses the slicing method to calculate the volume of a solid of revolution.

Media

Use an online integral calculator to learn more.

Example 6.7

Using the Slicing Method to find the Volume of a Solid of Revolution

Use the slicing method to find the volume of the solid of revolution bounded by the graphs of \(f \left(\right. x \left.\right) = x^{2} - 4 x + 5 , x = 1 , \text{and} x = 4 ,\) and rotated about the \(x -\text{axis} .\)

Solution

Using the problem-solving strategy, we first sketch the graph of the quadratic function over the interval \(\left[\right. 1 , 4 \left]\right.\) as shown in the following figure.

This figure is a graph of the parabola f(x)=x^2-4x+5. The parabola is the top of a shaded region above the x-axis. The region is bounded to the left by a line at x=1 and to the right by a line at x=4.

Figure 6.16 A region used to produce a solid of revolution.

Next, revolve the region around the x-axis, as shown in the following figure.

This figure has two graphs of the parabola f(x)=x^2-4x+5. The parabola is the top of a shaded region above the x-axis. The region is bounded to the left by a line at x=1 and to the right by a line at x=4. The first graph has a shaded solid below the parabola. This solid has been formed by rotating the parabola around the x-axis. The second graph is the same as the first, with the solid being rotated to show the solid.

Figure 6.17 Two views, (a) and (b), of the solid of revolution produced by revolving the region in Figure 6.16 about the \(x -\text{axis} .\)

Since the solid was formed by revolving the region around the \(x -\text{axis},\) the cross-sections are circles (step 1). The area of the cross-section, then, is the area of a circle, and the radius of the circle is given by \(f \left(\right. x \left.\right) .\) Use the formula for the area of the circle:

\[A \left(\right. x \left.\right) = \pi r^{2} = \pi \left[\right. f \left(\right. x \left.\right) \left]\right.^{2} = \pi \left(\right. x^{2} - 4 x + 5 \left.\right)^{2} (\text{step 2}) .\]

The volume, then, is (step 3)

\[\begin{aligned} V & = \int_{a}^{b} A \left(\right. x \left.\right) d x \\ & = \int_{1}^{4} \pi \left(\right. x^{2} - 4 x + 5 \left.\right)^{2} d x = \pi \int_{1}^{4} \left(\right. x^{4} - 8 x^{3} + 26 x^{2} - 40 x + 25 \left.\right) d x \\ & = \left(\pi \left(\right. \frac{x^{5}}{5} - 2 x^{4} + \frac{26 x^{3}}{3} - 20 x^{2} + 25 x \left.\right) \left|\right.\right)_{1}^{4} = \frac{78}{5} \pi . \end{aligned}\]

The volume is \(78 \pi / 5 .\)

Checkpoint 6.7

Use the method of slicing to find the volume of the solid of revolution formed by revolving the region between the graph of the function \(f \left(\right. x \left.\right) = 1 / x\) and the \(x -\text{axis}\) over the interval \(\left[\right. 1 , 2 \left]\right.\) around the \(x -\text{axis} .\) See the following figure.

This figure has two graphs. The first graph is the curve f(x)=1/x. It is a decreasing curve, above the x-axis in the first quadrant. The graph has a shaded region under the curve between x=1 and x=2. The second graph is the curve f(x)=1/x in the first quadrant. Also, underneath this graph, there is a solid between x=1 and x=2 that has been formed by rotating the region from the first graph around the x-axis.

This lesson is part of:

Applications of Integration

View Full Tutorial

Track Your Learning Progress

Sign in to unlock unlimited practice exams, tutorial practice quizzes, personalized weak area practice, AI study assistance with Lexi, and detailed performance analytics.