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.
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
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
The formula is
How 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 thus
and its equation is therefore
Now 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 by
Finally, 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.)
