Wikipedia:Reference desk/Archives/Mathematics/2016 June 6

= 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)
 * Use generalized spherical coordinates? See n-sphere. 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)