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The washer method

Some solids of revolution have cavities in the middle; they are not solid all the way to the axis of revolution. Sometimes, this is just a result of the way the region of revolution is shaped with respect to the axis of revolution. In other cases, cavities arise when the region of revolution is defined as the region between the graphs of two functions. A third way this can happen is when an axis of revolution other than the x -axis or y -axis is selected.

When the solid of revolution has a cavity in the middle, the slices used to approximate the volume are not disks, but washers (disks with holes in the center). For example, consider the region bounded above by the graph of the function f ( x ) = x and below by the graph of the function g ( x ) = 1 over the interval [ 1 , 4 ] . When this region is revolved around the x -axis, the result is a solid with a cavity in the middle, and the slices are washers. The graph of the function and a representative washer 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 is labeled “a” and has the two functions f(x)=squareroot(x) and g(x)=1 graphed in the first quadrant. f(x) is an increasing curve starting at the origin and g(x) is a horizontal line at y=1. The curves intersect at the ordered pair (1,1). In between the curves is a shaded rectangle with the bottom on g(x) and the top at f(x). The second graph labeled “b” is the same two curves as the first graph. The shaded rectangle between the curves from the first graph has been rotated around the x-axis to form an open disk or washer. The third graph labeled “a” has the same two curves as the first graph. There is a shaded region between the two curves between where they intersect and a line at x=4. The fourth graph is the same two curves as the first with the region from the third graph rotated around the x-axis forming a solid region with a hollow center. The hollow center is represented on the graph with broken horizontal lines at y=1 and y=-1.
(a) A thin rectangle in the region between two curves. (b) A representative disk formed by revolving the rectangle about the x -axis . (c) The region between the curves over the given interval. (d) The resulting solid of revolution.

The cross-sectional area, then, is the area of the outer circle less the area of the inner circle. In this case,

A ( x ) = π ( x ) 2 π ( 1 ) 2 = π ( x 1 ) .

Then the volume of the solid is

V = a b A ( x ) d x = 1 4 π ( x 1 ) d x = π [ x 2 2 x ] | 1 4 = 9 2 π units 3 .

Generalizing this process gives the washer method    .

Rule: the washer method

Suppose f ( x ) and g ( x ) are continuous, nonnegative functions such that f ( x ) g ( x ) over [ a , b ] . Let R denote the region bounded above by the graph of f ( x ) , below by the graph of g ( x ) , 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 ( g ( x ) ) 2 ] d x .

Using the washer method

Find the volume of a solid of revolution formed by revolving the region bounded above by the graph of f ( x ) = x and below by the graph of g ( x ) = 1 / x over the interval [ 1 , 4 ] around the x -axis .

The graphs of the functions and the solid of revolution are shown in the following figure.

This figure has two graphs. The first graph is labeled “a” and has the two curves f(x)=x and g(x)=1/x. They are graphed only in the first quadrant. f(x) is a diagonal line starting at the origin and g(x) is a decreasing curve with the y-axis as a vertical asymptote and the x-axis as a horizontal asymptote. The graphs intersect at (1,1). There is a shaded region between the graphs, bounded to the right by a line at x=4. The second graph is the same two curves. There is a solid formed by rotating the shaded region from the first graph around the x-axis.
(a) The region between the graphs of the functions f ( x ) = x and g ( x ) = 1 / x over the interval [ 1 , 4 ] . (b) Revolving the region about the x -axis generates a solid of revolution with a cavity in the middle.

We have

V = a b π [ ( f ( x ) ) 2 ( g ( x ) ) 2 ] d x = π 1 4 [ x 2 ( 1 x ) 2 ] d x = π [ x 3 3 + 1 x ] | 1 4 = 81 π 4 units 3 .
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Find the volume of a solid of revolution formed by revolving the region bounded by the graphs of f ( x ) = x and g ( x ) = 1 / x over the interval [ 1 , 3 ] around the x -axis .

10 π 3 units 3

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As with the disk method, we can also apply the washer method to solids of revolution that result from revolving a region around the y -axis. In this case, the following rule applies.

Rule: the washer method for solids of revolution around the y -axis

Suppose u ( y ) and v ( y ) are continuous, nonnegative functions such that v ( y ) u ( y ) for y [ c , d ] . Let Q denote the region bounded on the right by the graph of u ( y ) , on the left by the graph of v ( y ) , below by the line y = c , and above by the line y = d . Then, the volume of the solid of revolution formed by revolving Q around the y -axis is given by

V = c d π [ ( u ( y ) ) 2 ( v ( y ) ) 2 ] d y .

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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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