Finding the volume of a frustum

Question

I do know how to find this in a different way, I do not understand why this isn't working, however. Thank you for your time! The volume should be $\frac{\pi h}{3}*(r_2^2+r_2r_1+r_1^2)$ $$r_2-r_1\text{ corresponds to }\Delta z=h\text{ not }H$$

Answer 1

Volume by discs $$dV= \pi r^2 dz$$

$$ \frac{r}{z}= \tan \alpha$$

$$ dV= \pi r^2 d( r/ \tan \alpha)$$

$$ V=\int_{r_1}^{r_2}\pi r^2 d( r/ \tan \alpha) =\frac{\pi}{3} (r_2^3-r_1^3)/\tan \alpha$$re where $$\tan \alpha= \frac{r_2-r_1}{h}$$ Plug in and result follows: $$ V=\frac{\pi h}{3}(r_1^2+r_2^2+r_1 r_2)$$