Can anyone see the connection between the Gaussian curvature of the ellipsoid $x^2+y^2+az^2-1=0$ where $a>0$ and the integral $\int_0^1 {1\over (1+(a-1)w^2)^{3\over 2}}dw$?
I am guessing Gauss Bonnet, since that is what the related chapter is about. Is there a quick / clever way of finding the curvature of this surface?