Volume__The_Disk_Method - You have already learned that area is only one of the many applications of the definite integral Another important application

Volume__The_Disk_Method - You have already learned that...

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volution You have already learned that area is only one of the many applications of the definite integral. Another important application is its use in finding the volume of a three- dimensional solid. In this section you will study a particular type of three-dimensional solid--one whose cross sections are similar. Solids of revolution are used commonly in engineering and manufacturing. Some examples are axles, funnels, pills, bottles, and pistons, as shown in Figure 7.12. Solids of revolution Figure 7.12 If a region in the plane is revolved about a line, the resulting solid is a solid of revolution, and the line is called the axis of revolution. The simplest such solid is a right circular cylinder or disk, which is formed by revolving a rectangle about an axis adjacent to one side of the rectangle, as shown in Figure 7.13. The volume of such a disk is Volume of disk -- (area of disk)(width of disk) = 7rR2w where R is the radius of the disk and w is the width. To see how to use the volume of a disk to find the volume of a general solid of revolution, consider a solid of revolution formed by revolving the plane region in Figure 7.14 about the indicated axis. To determine the volume of this solid, consider a representative rectangle in the plane region. When this rectangle is revolved about the axis of revolution, it generates a representative disk whose volume is AV = "rrR2Ax. Approximating the volume of the solid by n such disks of width Ax and radius R(xi) produces Volume of solid ~ 2 7fiR(x; )]2 Ax i=1 I1 i=l ~In Figure 7.15, note that you can determine the variable of integration by placing a representative rectangle in the plane region "perpendicular" to the axis of revolution. If the width of the

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