numerical integration of surface integral

I hope this is not a silly Q. I have an integral expressed as:

where and is a scalar function (einstein summation over 1,2,3). The 3D version looks more like: and . I am assuming a simple trapz twice won't be correct to implement here, and I am not sure how to proceed. This looks like a surface/volume integral, and I am confused how best to implement this in matlab as this is discrete data so I need to carry out numerical integration. I would appreciate any ideas here

[Torsten]

... as this is discrete data ...

Discrete data for what ? How are they given ? What are the surfaces/volumes you have to integrate over ?

[Uche Agbogwu]

I am sorry for the late response. I had expected to get notifications from matlab when there's a respinse, but did not. I have the spatial autocorrelations of velocity from a large eddy simulation computation, so that and . So that the length Lcomes from the the integration over (what seems to be) .

Answer 1

[Uche Agbogwu]

I decided to convert the integrals to polar coordinates. That way, vanishes and all that remains is the integral . In 2D, that becomes;

and in 3D,

that way I can easily integrate.