The Disk MethodAnother important application of the definite integral is its use in finding thevolume of a three-dimensional solid. In this section, you will study a particulartype of three-dimensional solid—one whose cross sections are similar. You willbegin 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 revolutionis formed by revolving aplane 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, considera continuous function fthat is nonnegative on the interval Suppose that thearea of the region is approximated by nrectangles, each of width as shown inFigure 5.26. By revolving the rectangles about the x-axis, you obtain ncirculardisks, each with a volume of The volume of the solid formed byrevolving the region about the x-axis is approximately equal to the sum of thevolumes of the ndisks. Moreover, by taking the limit as napproaches infinity, youcan see that the exact volume is given by a definite integral. This result is calledthe Disk Method.f xi2x.x,a, b.SECTION 5.7Volumes of Solids of Revolution3715.7V 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 MethodThe volume of the solid formed by revolving the region bounded by thegraph of and the axis about the axis isVolumebaf x2dx.x-a≤x≤bx-fR∆xx = bx = aPlane regionRepresentativerectangleSolid ofrevolutionveAxis ofrevolution∆xApproximationby ndisksRepresentatidiskApproximation by nrectanglesFIGURE 5.26Axis of revolutionregionPlaneFIGURE 5.25
372CHAPTER 5Integration and Its ApplicationsE X A M P L E 1Finding the Volume of a Solid of RevolutionFind the volume of the solid formed by revolving the region bounded by the graphof and the x-axis about the x-axis.SOLUTIONBegin by sketching the region bounded by the graph of and the axis. As shown in Figure 5.27(a), sketch a representative rectangle whoseheight is and whose width is From this rectangle, you can see that theradius of the solid isUsing the Disk Method, you can find the volume of the solid of revolution.Disk MethodSubstitute for Expand integrand.Find antiderivative.