91Ó°ÊÓ

The sides of the rectangle are 2a,2b.

The volumes of the cone and cylinder are straightforward, and the volume of the ellipsoid can be found by transforming into a system where it turns into a sphere.

The volume of the cylinder is as follows:

$$\begin{gathered} {\mathrm{V}}_1=\left({\mathrm{Ï„Ï„²¹}}^2\right)\left(2\mathrm{b}\right) \\ =2{\mathrm{Ï„Ï„²¹}}^2\mathrm{b} \end{gathered}$$

Find the volume of the ellipsoid and transform it as follows:

$$\begin{gathered} x=x\text{'}a \\ y=y\text{'}b \\ z=z\text{'}b \end{gathered}$$

$$\begin{gathered} {\mathrm{V}}_2={∫}_{\mathrm{x}}{∫}_{\mathrm{y}}{∫}_{\mathrm{z}}\mathrm{dx}\text{'}\mathrm{dy}\text{'}\mathrm{dz}\text{'}{\mathrm{a}}^2\mathrm{b} \\ ={\mathrm{a}}^2\mathrm{b}\frac{4\mathrm{ττ}}{3} \\ =\frac{4\mathrm{ττ}}{3}{\mathrm{a}}^2\mathrm{b} \end{gathered}$$

Where in the transformed equation ellipsoid turns into a sphere of radius 1.

The volume of the cone can also be found by transforming:

$$\begin{gathered} {\mathrm{V}}_3={\mathrm{a}}^2\mathrm{b}{∫}_{\mathrm{o}}^{2\mathrm{Ï„Ï„}}\mathrm{»åθ}{∫}_{-1}^1\mathrm{dz}\text{'}{∫}_0^1\mathrm{rdr} \\ =2{\mathrm{Ï„Ï„²¹}}^2\mathrm{b}{∫}_{-1}^1\mathrm{dz}\frac{1}{2}{\mathrm{z}}^2 \\ =\frac{2\mathrm{Ï„Ï„}}{3}{\mathrm{a}}^2\mathrm{b} \end{gathered}$$

Origin at the center of the ellipsoid.

Thus, proving the theorem results in the following,

$$\begin{gathered} \frac{V_2}{V_3}=2 \\ \frac{V_1}{V_3}=3 \end{gathered}$$

Hence, the volumes are in the ratio 1:2:3 .