As far as I know any figure shaped like a cone (pyramid, cones, etc.) has a volume equal to one third of the base area times the height, so I planned on proving this statemente for any figure with such qualities using calculus, but i'm stuck an wish some help (please no full answer).
This is what I've thought of up until now. Say $A(x)$ is a function of the area of a transversal cut in terms of the height. Therefore $A(h)=0$ and $A(0)$ is the area of the base or equivalently $A(h)$ is the area of the base and $A(0)=0$ (I'll use the latter). Using Cavalieri's Principle we determine that the volume of the cone-like figure is given by adding the areas of all the transversal cuts, or in notation: $\int_0^h A(x)\, dx$
I also know there must be a rate of change that relates $A(x)$ to $x$, and this is the part where I'm stuck. :(
Thanks in advance.