>>149563399∫ ∫ 2πr dr dx for x from 0 to h for r from 0 to xR/h
∫ [(2π(xR/h)2/2) - (2π02/2)] dx for x from 0 to h
∫ πR2x2/(h2) dx for x from 0 to h
πR2h3/(3h2) - πR203/(3h2)
πR2h/3
Where R is the radius at the base and h is the height of the cone.
Radius at a given height of the cone is rR/h because of similar triangles (h/R = x/r)
2πr is of course the circumference at a given radius.