Concerning [an integral] and a Taylor series method

33. R. S. Anderssen, R. P. Brent, D. J. Daley and P. A. P. Moran, Concerning $\int_0^1\cdots\int_0^1({x_1}^2+\cdots+{x_k}^2)^{1/2}dx_1\ldots dx_k$ and a Taylor series method, SIAM J. Applied Mathematics 30 (1976), 22-30. MR 52#15773.

Abstract: dvi (3K), pdf (81K).

Paper: pdf (648K).


The integral of the title equals the mean distance mk from the origin of a point uniformly distributed over the k-dimensional unit hypercube Ik. Closed form expressions are given for k = 1, 2 and 3, while for general k, mk is asymptotic to (k/3)1/2. Using inter alia methods from geometry, Cauchy-Schwarz inequalities and Taylor series expansions, several inequalities and an asymptotic series for mk are established. The Taylor series method also yields a slowly converging infinite series for mk and can be applied to more general problems including the mean distance between two points independently distributed at random in Ik.


For recent references, see the MathWorld "Hypercube Line Picking" page.

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