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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?

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I don't know, but you might find it useful that the integral of the Gaussian curvature over all the ellipsoid, according to Gauss Bonnet, is $4\pi$ –  yohBS Mar 11 '12 at 17:45
    
@yohBS Isn't this kind of obvious for ellipsoids? And I still don't see the connection to the integral, which works out as $\frac{1}{\sqrt{a}}$ . –  draks ... Apr 5 '12 at 22:56
    
I don't know about quick/clever, but the surface is given by $$z=a^{-1/2}\sqrt{1-x^2-y^2}$$ and the Gaussian curvature is given by $$K=\frac{z_{xx} z_{yy} -z_{xy}^2}{(1+z_x^2+z_y^2)^2}=\frac{a}{\left(a+r^2-a r^2\right)^2}$$ where $r=\sqrt{x^2+y^2}$. The total curvature is given by integrating over $r$ but you need to multiply by the surface element $\sqrt{1+z_x^2+z_y^2}$. This results in $$4\pi \sqrt{a}\underset{0}{\overset{1}{\int }}\frac{r}{ \left(a+(1-a)r^2\right)^{3/2}\sqrt{\left(r^2-1\right)}}dr=4\pi$$ whish is somewhat similar to the integral you proposed. Hope that helps. –  yohBS Apr 8 '12 at 13:16
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