SECTION 5.7 5.7 Volumes of Solids of Revolution VOLUMES OF SOLIDS OF R - SECTION 5.7 5.7 Volumes of Solids of Revolution VOLUMES OF SOLIDS OF REVOLUTION

SECTION 5.7 5.7 Volumes of Solids of Revolution VOLUMES OF SOLIDS OF R

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The Disk Method Another important application of the definite integral 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. You will begin with solids of revolution. These solids, such as axles, funnels, pills, bottles, and pistons, are used commonly in engineering and manufacturing. As shown in Figure 5.25, a solid of revolution is formed by revolving a plane region about a line. The line is called the axis of revolution. To develop a formula for finding the volume of a solid of revolution, consider a continuous function f that is nonnegative on the interval Suppose that the area of the region is approximated by n rectangles, each of width as shown in Figure 5.26. By revolving the rectangles about the x -axis, you obtain n circular disks, each with a volume of The volume of the solid formed by revolving the region about the x -axis is approximately equal to the sum of the volumes of the n disks. Moreover, by taking the limit as n approaches infinity, you can see that the exact volume is given by a definite integral. This result is called the Disk Method. f x i 2 x . x , a , b . SECTION 5.7 Volumes of Solids of Revolution 371 5.7 V O L U M E S O F S O L I D S O F R E V O L U T I O N Use the Disk Method to find volumes of solids of revolution. Use the Washer Method to find volumes of solids of revolution with holes. Use solids of revolution to solve real-life problems. The Disk Method The volume of the solid formed by revolving the region bounded by the graph of and the axis about the axis is Volume b a f x 2 dx . x - a x b x - f R x x = b x = a Plane region Representative rectangle Solid of revolution ve Axis of revolution x Approximation by n disks Representati disk Approximation by n rectangles FIGURE 5.26 Axis of revolution region Plane FIGURE 5.25
372 CHAPTER 5 Integration and Its Applications E X A M P L E 1 Finding the Volume of a Solid of Revolution Find the volume of the solid formed by revolving the region bounded by the graph of and the x -axis about the x -axis. SOLUTION Begin by sketching the region bounded by the graph of and the axis. As shown in Figure 5.27(a), sketch a representative rectangle whose height is and whose width is From this rectangle, you can see that the radius of the solid is Using the Disk Method, you can find the volume of the solid of revolution. Disk Method Substitute for Expand integrand. Find antiderivative.

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