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.
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,
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.