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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.

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Try the matlab function 'triplequad'. – Peter Sheldrick Jun 27 '12 at 12:13
@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 $r_{34}$. – Isaac Jun 27 '12 at 14:27

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