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June 6

Expected distance between point inside a hypercube and a vertex

Let $I_{n}=\int _{0}^{1}\int _{0}^{1}...\int _{0}^{1}{\sqrt {x_{1}^{2}+x_{2}^{2}+...x_{n}^{2}}}d{x_{1}}...d{x_{n}}$ . $I_{1}=1/2,I_{2}=({\sqrt {2}}+\sinh ^{-1}{1})/3,I_{3}\approx 0.960591956455053$ . What would be the best way to numerically evaluate this integral in higher dimensions? Or, if there's a simple closed form what is it? 24.255.17.182 (talk) 22:01, 6 June 2016 (UTC)[reply]

Use generalized spherical coordinates? See n-sphere#Spherical coordinates. Not that this is gonna make things much easier though - I'm sure that a hypercube's boundaries can be expressed as trigonometric functions in spherical coordinates, and integrating trigonometric functions can always be done symbolically. For example, using the notation from that article, we could write

x_{1}=1

as

r=\sec(\phi _{1})

.--Jasper Deng (talk) 00:18, 7 June 2016 (UTC)[reply]