Numerically integrating with weight factors

Hi, i am integrating a stiffness matrix over two euler angles, phi, theta.

if i integrate the stiffness matrix over both angles, and then divide by the area of a unit sphere, it is treated as if each particular direction is weighted equally, and for a transversely isotropic material, when integrated will produce an isotropic material.

I am interested in biasing the integral however with a separate probability distribution (which i have at the moment as a gaussian curve revolved around an axis, in meshgrid form).

How do i specify when integrating - that i want the particular numerical integration function (say quad) to while integrating at a particular point phi,theta - to go and grab the weight factor from my table?

Is there any way to do this?

any help would be greatly appreciated.