Volume of a curved cone

Question

Apparently, the volume of this cone is $\frac{1}{16}\pi r^2h$. My question is why this is the case, can someone please geometrically explain the reason behind the $\frac{1}{16}$ bit. Thanks.

Answer 1

Given variation of radius is second degree.

$$ (z/h)^2 = (y/r) $$

$$ V = \pi \int_0^h y^2 dz $$

Evaluating it

$$ V = \frac{\pi r^2 h} {5} $$

The given degree of " cone" parabola is too low.

If it is given as

$$ (z/h)^{15/2} = (y/r) $$

only then given result is ok.