Why average area of the horizontal slices of the conical frustum doesn't work for it's volume?

I would like to react to one of the answers on this thread (I don't have enough rep to make a comment):

Use cylinder's formula for frustum (conical frustum)

Where is answered:

Essentially, what you'd need is the average of the areas of the horizontal slices into which the frustrum is cut by planes paralell to its base, not their diameters.

I would like to ask if it is really working, I tried it with a conical frustum of r1 = 4, r2 = 2, h = 10 and a get result of V =100$\pi$ , the result give by formula $\frac1 3\pi h(r^2+R^2+rR)$ is $293.215$

Did I something wrong in the calculation or is the idea of "transforming" a cone frustum into cylinder with base area equal to an average area of two frustum's slices wrong ?

Thank you for your answers.

Solution 1:

While the idea of averaging the areas is correct, the problem here is that it is not a linear average, since the area doesn't change linearly with the height. The correct average is obtained by integration:

$$ \int_0^{h}\pi\left(r_1 - \frac{(r_1-r_2)}{h}r\right)^2 \;\mathrm{d}r. $$

This averages the area function where the radius changes linearly between $r_1$ and $r_2$ with $h$, and integrates to the formula you gave.

Since the answer to the other question didn't specify what kind of average, it is technically not wrong, just misleading.