In addition to finding the volume of unusual shapes, integration can help you to derive volume formulas. For example, you can use the disk/washer method of integration to derive the formula for the volume of a cone.
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(calculate volume of a truncated cone) Definition of a frustum of a right circular cone: A frustum of a right circular cone (a truncated cone) is a geometrical figure that is created from a right circular cone by cutting off the tip of the cone perpendicular to its height H. The small h is the height of the truncated cone. The volume of the waffle cone with a circular base with radius 1.5 in and height 5 in can be computed using the equation below: volume = 1/3 × π × 1.5 2 × 5 = 11.781 in 3. Bea also calculates the volume of the sugar cone and finds that the difference is.
Integration works by cutting something up into an infinite number of infinitesimal pieces and then adding the pieces up to compute the total. The disk/washer method cuts up a given shape into thin, flat disks or washers; this makes it useful for shapes with circular cross-sections, like, well, cones.
The following practice question asks you to apply the disk method for just this purpose.
Practice question
![Volume Volume](/uploads/1/2/6/3/126322292/347261221.bmp)
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Use the disk method to derive the formula for the volume of a cone. Hint: What’s your function? See the following figure. Your formula should be in terms of r and h.
Answer and explanation
![Volume Of A Cone Volume Of A Cone](http://rlv.zcache.com/volume_of_a_cone_postcards-r956f0e665a064514a66cc4f025fffb8e_vgbaq_8byvr_512.jpg)
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The formula isHow do you get it? First, find the function that revolves about the x-axis to generate the cone.The function is the line that goes through (0, 0) and (h, r). Its slope is thusand its equation is thereforeNow express the volume of a representative disk. The radius of your representative disk is f (x) and its thickness is dx. Thus, its volume is given byFinally, add up the disks from x = 0 to x = h by integrating. (Don’t forget that r and h are constants that behave like numbers.)
![](/uploads/1/2/6/3/126322292/480975903.jpg)