A box integral for dimension 
with parameters 
and 
is defined as the expectation of distance from a fixed point 
of a point 
chosen at random over the unit 
-cube,
![X_n(s,q)
=int_0^1...int_0^1_()_(n)[(r_1-q_1)^2+...+(r_n-q_n)^2]^(s/2)dr_1...dr_n](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FBoxIntegral%2FNumberedEquation1.svg) |
(1) |
(Bailey et al. 2006).
Two special cases include
which, with 
,
correspond to hypercube point picking (to
a fixed vertex) and hypercube line picking,
respectively.
Hypercube point picking to the center is given by
 |
(4) |
See also
Hypercube Line Picking, Hypercube Point Picking, Unit Cube, Unit Square, Unit Square Integral
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References
Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. "Box Integrals." Preprint. Apr. 3, 2006.Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, pp. 238 and 272, 2007.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Box Integral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BoxIntegral.html