Suppose the ellipse has equation $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$. 
I understand the way to obtain the surface area of the ellipsoid is to rotate the curve around y-axis and use surface of revolution. 

But I'm wondering how to obtain this formula that I found on the internet $Area=2\pi b^{2}\left ( 1+\frac{a}{b} \frac{arcsin\varepsilon }{\varepsilon }\right )$ with with eccentricity $\varepsilon$ given by $\varepsilon =\sqrt{1-\left ( \frac{b}{a} \right )^{2}}$

Can anybody help prove it for me just from the ellipse equation above and surface of revolution? Thanks a lot:).