$$\begin{aligned} & E_1=\frac{x^2}{9}+\frac{y^2}{4} \Rightarrow e=\sqrt{1-\frac{4}{9}}=\frac{\sqrt{5}}{3} \\ & E_2: \frac{x^2}{a^2}+\frac{y^2}{4}=1 \\ & e=\frac{\sqrt{5}}{3}=\sqrt{1-\frac{a^2}{4}} \Rightarrow \frac{5}{9}=1-\frac{a^2}{4} \\ & a^2=\frac{16}{9} \\ & E_2: \frac{x^2}{16}+\frac{y^2}{4}=1 \\ & E_3: \frac{x^2}{\frac{16}{9}}+\frac{y^2}{b^2}=1 \\ & e=\frac{\sqrt{5}}{3}=\sqrt{1-\frac{b^2}{16}} \Rightarrow b^2=\frac{64}{81} \end{aligned}$$

$$\begin{aligned} & \mathrm{E}_3=\frac{\mathrm{x}^2}{\frac{16}{9}}+\frac{\mathrm{y}^2}{\frac{64}{81}}=1 \\ & \mathrm{~A}_1=\pi \times 3 \times 2 \Rightarrow 6 \pi \\ & \mathrm{~A}_2=\pi \times \frac{4}{3} \times 2=\frac{8 \pi}{3} \\ & \mathrm{~A}_3=\pi \times \frac{4}{3} \times \frac{8}{9}=\frac{32 \pi}{81} \\ & \sum_{\mathrm{i}=1}^{\infty} \mathrm{A}_{\mathrm{i}}=6 \pi+\frac{8 \pi}{3}+\frac{32 \pi}{81}+\ldots \infty \Rightarrow \frac{6 \pi}{1-\frac{4}{9}} \Rightarrow \frac{54 \pi}{5} \\ & \therefore \frac{5}{\pi} \sum_{\mathrm{i}=1}^{\infty} \mathrm{A}_{\mathrm{i}} \Rightarrow \frac{5}{\pi} \times \frac{54 \pi}{5}=54 \end{aligned}$$