Finding corrections to volume of a two-sphere in 3-sphere geometry

[Raziel2701]

Homework Statement

Given the volume of a 2-sphere in a 3-sphere geometry,
$$V=4\pi a^3[\frac{1}{2}\arcsin (r/a) -\frac{r}{2a}[1-(\frac{r}{a})^2]^{(1/2)}]$$, derive the 1st order correction when r/a<<1

I am also given that this is approximately $$V\approx\frac{4\pi r^3}{3}[1+()]$$

where inside the blank parenthesis should read "corrections of order (r/a)^2

The Attempt at a Solution

So since r/a<<1 I figure that the only important term should be the arcsine and that I should expand it, but if I neglect everything but the arcsine I don't get the approximate expression, I just get the 4pir^3 times the first few terms of the arcsine expansion, not to mention that I am not entirely sure what is meant by corrections of order (r/a)^2 since the arcsine expansion has odd powers.

 

[tiny-tim]

Hi Raziel2701!

(have a pi: π and a square-root: √ and try using the X² icon just above the Reply box )

So since r/a<<1 I figure that the only important term should be the arcsine and that I should expand it, but if I neglect everything but the arcsine I don't get the approximate expression, I just get the 4pir^3 times the first few terms of the arcsine expansion, not to mention that I am not entirely sure what is meant by corrections of order (r/a)^2 since the arcsine expansion has odd powers.

But arcsine(r/a) is approximately r/a, so it's almost the same size as the other part …

you need to expand both of them