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 thusand 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 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.)