Write an integral expression for the volume of a cone with height *h* and base radius *r*. Evaluate your expression to get a formula for the volume of a cone.

Answer

This is just like the example except there aren't any numbers. Since *h* is being used for the height of the cone, let's use *y* for the distance from the tip of the cone to the slice. Then the thickness of the slice is Δ *y*. We cut the cone down the middle to see the similar triangles. The radius of the slice is *x*. Using similar triangles,

so

The circular side of the slice has area

and the slice has volume

Since the variable *y* goes from 0 (at the tip of the cone) to *h* (at the base of the cone), the volume of the cone is

Let's work out the integral. Remember that *r*, *h*, and π are constants so we can pull them out in front.