# Computing ${\int\limits_{0}^{\chi}\int\limits_{0}^{\chi}\int\limits_{0}^{2\pi}\sqrt{1- \cdots} \, d\phi \, d\theta_1 \, d\theta_2}$?

This question led me to this integral I can not solve:

$${\int\limits_{0}^{\chi}\int\limits_{0}^{\chi}\int\limits_{0}^{2\pi}\sqrt{1-\left(\sin\theta_{1}\sin\theta_{2}\cos\phi+\cos\theta_{1}\cos\theta_{2}\right)^{2}} \sin\theta_1 \sin\theta_2 \,\mathrm d\phi \,\mathrm d\theta_1 \,\mathrm d\theta_2}$$

where ${\chi \in [0,\frac{\pi}{2})}$

Any help would be appreciated!

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Thank you! There was definitely a mistake. But I expect both $\theta_1$ and $\theta_2$ to be in $[0,\chi]$. The point is that it's a cone, not a sphere. –  Installero May 21 '12 at 8:58
This is the integral over the sine of the angle between two points on the spherical cap $\theta\in[0,\chi]$ with respect to the natural measure on the unit sphere. –  joriki May 23 '12 at 7:54
Maybe the integral does not have any simple closed expression. This is in contrast to the average value of cosine, which is remarkably simple (I got $\cos^4(\chi/2)$ for the cosine). Perhaps you should take a broader view of this problem. Hypothetically speaking, if you had a "closed expression" for this integral that involved elliptic integrals and was a couple of pages long -- what would you do with it? –  user31373 May 24 '12 at 22:02