Asymptotic analysis of Integrals of powers of sine and their application to intersections of hyperspheres

I am trying to estimate the probability of an event in an algorithm. For simplicity, assume there are two hyperspheres of radius $r$, at a distance $r$ from each other. I am looking to see how the volume of the intersection of the hyperspheres compares to the volume of a single hypersphere as dimensionality increases.

I consulted wikipedia here: https://en.wikipedia.org/wiki/Spherical_cap And here: http://en.wikipedia.org/wiki/Volume_of_an_n-ball

And have found the ratio $V_{intersect}/V_{hypersphere}$ reduces to analyzing the asymptotic value of: $f(d) = \int_0^{\pi/3} sin^d(t)dt$.

As $d \to \infty$, $f(d)$ goes to $0$. Futhermore, it seems like this is going to be $O(\frac{1}{2^d})$, since the $d$ is in the exponent of the above integral.

This is slightly disappointing and I'm wondering if my analysis checks out. Can anyone offer their thoughts?

• The Laplace method can be used to find that $$f(d) \sim \left(\frac{\sqrt{3}}{2}\right)^d \int_{-\infty}^{0} e^{dt/\sqrt{3}}\,\mathrm dt = \frac{\sqrt{3}}{d} \left(\frac{\sqrt{3}}{2}\right)^d$$ as $d \to \infty$. – Antonio Vargas May 10 '15 at 7:31
• I don't see how to do it, but thank you! This follows my understanding that it is exponential in $d$. – Bryce Sandlund May 12 '15 at 4:07