How to evaluate this integral analytically or numerically:

$$ I=\iiint dr_{12}dr_{13}dr_{14} $$

constrained by

$$ r_{12}<r_0,\\r_{23}<r_0,\\r_{34}<r_0, $$

where $r_0$ is a given real number and $$r_{ij}=\sqrt{(x_i-x_j)^2+(y_i-y_j)^2+(z_i-z_j)^2}$$

I have been that this integral cannot be calculated analytically and one way for the numerical calculation is to use Monte Carlo methods, but I don't know how to implement this in MATLAB.

  • $\begingroup$ @PeterSheldrick: Thanks. How should I give the constraints to this function? Note that the differentials are $dr_{12}$, $dr_{13}$, and $dr_{14}$, while the constraints are imposed on $r_{12}$, $r_{23}$, and $r_{34}$. $\endgroup$
    – Isaac
    Jun 27, 2012 at 14:27


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