Most who have been to college recognizes the red glasses used for beerpong and alcohol consumption. My question is about why one method is better at estimating the volume and why.
The exact volume of a frostum with a small radius $r$, larger radius $R$ and $h$ is given as
$$ V = \frac{\pi}{3} ( R^2 + rR + r^2 ) h $$
One method to estimate the volume of the frustum is to take the average volume of a cylinder with radius $R$ and $r$. Eg
$$ V_1 = \frac{1}{2}\left[ \pi R^2 h + \pi r^2 h \right] = \frac{\pi}{2} \left( R^2 + r^2 \right) h $$
Another method is to estimate the volume by taking the averages of the radii. So $\tau = (R + r)/2$.
$$ V_2 = \pi \tau^2 h = \pi \left( \frac{R+r}{2} \right)^2h = \frac{\pi}{4} \left( R^2 + 2rR + r^2 \right) h $$
By comparison one now obtains
$$ V - V_1 = -\frac{\pi}{6}(R-r)^2h \quad \text{and} \quad V - V_2 = \frac{\pi}{12}(R-r)^2h $$
Now I have two questions.
1: Why is $V_2$ a better estimate of the volume of the frustum, and why is $V_1 > V > V_2$?
2: Why is $V_2$ exactly twice as good as V_1? Eg $|V - V_1| = 2|V - V_2|$