The surface area of an oblate ellipsoid

[Adams2020]

TL;DR Summary

How to get the surface area of an oblate ellipsoid in terms of ε?

In "An Introduction to Nuclear Physics by W. N. Cottingham, D. A. Greenwood" for the surface area of an oblate ellipsoid, the following equation is written for small values of ε :

The book has said this without proof.
I found the following formula for the desired shape:

No matter how hard I tried, I could not get the result of the book with this formula. I used Taylor's expansions in the calculation, but unfortunately my result does not match the book. Help me, please!

 

[hutchphd]

What did you get...it certainly works through order zero.

 

[Adams2020]

What did you get...it certainly works through order zero.

My final result is as follows:
4*pi*R^2 (1+ε+ (7/5)ε ^2+ (36/35)ε ^3 +...)
Which has a bad difference with the result of the book.

 

[hutchphd]

There is no term linear in $\epsilon$ because the function is even...remember $\epsilon=\frac c a$. How did they define R in the first approximation?

 

[Adams2020]

There is no term linear in $\epsilon$ because the function is even...remember $\epsilon=\frac c a$. How did they define R in the first approximation?

My data is as follows:

 

[hutchphd]

What then is c in your formula 2??

 

[Adams2020]

What then is c in your formula 2??

A similar formula was stated by our nuclear professor, but with the difference that there was b instead of c. So I thought in this case b = c

 

[hutchphd]

I suggest double checking the definitions eccentricities (see wikipedia) and if you wish to show your complete calculation (all steps) I can perhaps help. Otherwise there are too many mystery definitions...certainly these should agree through order $\epsilon^2$

 

[Adams2020]

I suggest double checking the definitions eccentricities (see wikipedia) and if you wish to show your complete calculation (all steps) I can perhaps help. Otherwise there are too many mystery definitions...certainly these should agree through order $\epsilon^2$

ok thanks