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| Subject keyword(s) | National Curve Bank |
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| Resource type | Instructional Material |
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Back to . . . . Curve Bank Home Dr. Lou Talman Metro State Dept. of Mathematical and Computer Sciences http://clem.mscd.edu/~talmanl Application of the Definite Integral Volumeof a Solid of Revolution NCB Deposit # 36 Calculus of Volumes Revolvinga plane figure about an axis generates a volume. Definition: Consider theregionbetween the graph of a continuous function y = f(x)and the x-axis from x = a to x= b. Planeregion Replaythe animation Anotherexample . . . Replaythe animation Thefunction need not be one of the standard plane figures found inelementarygeometry. Theperpendicular cross-section, or slice, is still a circle. Another version. . . Replaythe animation Revolutionabout the y- axis: For more of Dr. Talman'sanimationssee http://clem.mscd.edu/~talmanl/MathAnim.html Replay theanimation Note: If the cross-sectionis NOT a disk, but a washer, we first write the area ofthewasher by subtracting the area of the inner cross-section from the areaofthe outer cross-section. Then we set up the integral, beingcareful to choose whether the rotation is about x- or y-. Background for the student. . . . Significance ofVolumesand Surfaces Thedefiniteintegral is an amazingly versatile tool. In Deposit #36 we seehowa rotated plane figure sweeps out a volume. But the process of visualizing this one concepthas far wider applications. We can compute area, volume, arclengthand surface area using essentially the same mental process. First wedividean object into smaller pieces - n smaller pieces of athicknessthat will eventually become our dx or dy. Weapproximatea quantity for each of the small pieces. This is usually an areaora length. We add up the approximations and then take a limit. Thus,we have intuitively derived a definite integral. Sketch the solid and a typical crosssection. Find a formula for the crioss-sectionalarea A(x). Find the limits for integration on therotational axis. Integrate A(x) to findthe volume. A Famous Paradox Gabriel's Horn or Torricelli's Trumpet If the function y = 1/x is revolved around the x-axis for x > 1, the figure has a finite volume,but infinite surface area. Gabriel'sHorn or Torricelli's Trumpet Another example: Note that y in the equationhas only the first power and becomes the axis of rotation for thisellipticalparaboloid. Volume of a Circular Paraboloid Printed References Modern calculus texts will have extensivematerial on volume of solids of revolution in the chapter on definiteintegrals. James Stewart, Calculus, 5th ed.,THOMSONBrooks/Cole,2003, p. 382. Howard Anton, Calculus, 6th ed., JohnWileyand Sons, 1999, p. 468. Finney, Weir, and Giordano, Thomas' Calculus, 10th ed., Addison-Wesley, 2001, p. 415. Smith and Minton, Calculus, 2nd ed.,McGraw-Hill,2002, p. 411. For Mathematica® code thatwillcreate many variations of these graphs see Gray, A., MODERNDIFFERENTIALGEOMETRY of Curves and Surfaces with Mathematica®, 2nd.ed.,CRC Press, 1998. Applications CAT scans MRIs Industrial designs Containers and packaging Construction Weight of a part turned on a lathe.