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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.

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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 ( x ) = x 2 4 x + 5 , x = 1 , and x = 4 , and rotated about the x -axis .

Using the problem-solving strategy, we first sketch the graph of the quadratic function over the interval [ 1 , 4 ] 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.
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.
Two views, (a) and (b), of the solid of revolution produced by revolving the region in [link] about the x -axis .

Since the solid was formed by revolving the region around the x -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 ( x ) . Use the formula for the area of the circle:

A ( x ) = π r 2 = π [ f ( x ) ] 2 = π ( x 2 4 x + 5 ) 2 (step 2) .

The volume, then, is (step 3)

V = a h A ( x ) d x = 1 4 π ( x 2 4 x + 5 ) 2 d x = π 1 4 ( x 4 8 x 3 + 26 x 2 40 x + 25 ) d x = π ( x 5 5 2 x 4 + 26 x 3 3 20 x 2 + 25 x ) | 1 4 = 78 5 π .

The volume is 78 π / 5 .

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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 ( x ) = 1 / x and the x -axis over the interval [ 1 , 2 ] around the x -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.

π 2

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The disk method

When we use the slicing method with solids of revolution, it is often called the disk method    because, for solids of revolution, the slices used to over approximate the volume of the solid are disks. To see this, consider the solid of revolution generated by revolving the region between the graph of the function f ( x ) = ( x 1 ) 2 + 1 and the x -axis over the interval [ −1 , 3 ] around the x -axis . The graph of the function and a representative disk are shown in [link] (a) and (b). The region of revolution and the resulting solid are shown in [link] (c) and (d).

This figure has four graphs. The first graph, labeled “a” is a parabola f(x)=(x-1)^2+1. The curve is above the x-axis and intersects the y-axis at y=2. Under the curve in the first quadrant is a vertical rectangle starting at the x-axis and stopping at the curve. The second graph, labeled “b” is the same parabola as in the first graph. The rectangle under the parabola from the first graph has been rotated around the x-axis forming a solid disk. The third graph labeled “c” is the same parabola as the first graph. There is a shaded region bounded above by the parabola, to the left by the line x=-1 and to the right by the line x=3, and below by the x-axis. The fourth graph labeled “d” is the same parabola as the first graph. The region from the third graph has been revolved around the x-axis to form a solid.
(a) A thin rectangle for approximating the area under a curve. (b) A representative disk formed by revolving the rectangle about the x -axis . (c) The region under the curve is revolved about the x -axis , resulting in (d) the solid of revolution.

We already used the formal Riemann sum development of the volume formula when we developed the slicing method. We know that

V = a b A ( x ) d x .

The only difference with the disk method is that we know the formula for the cross-sectional area ahead of time; it is the area of a circle. This gives the following rule.

Rule: the disk method

Let f ( x ) be continuous and nonnegative. Define R as the region bounded above by the graph of f ( x ) , below by the x -axis, on the left by the line x = a , and on the right by the line x = b . Then, the volume of the solid of revolution formed by revolving R around the x -axis is given by

V = a b π [ f ( x ) ] 2 d x .

The volume of the solid we have been studying ( [link] ) is given by

Practice Key Terms 5

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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