i am having an issue finding the variational average of a stiffness matrix. The stiffness matrix is of a unidirectional composite, and a function of phi, theta. I have some cut-up data that shows that the fibre orientation follows a gaussian curve centered about 22 degrees (phi direction), theta is axisymmetric. Therefore, the probability density function has phi on the x-axis, and f(phi) on the y-axis. Therefore, i have a probability density line that i am using to integrate over a sphere using the following (over a hemisphere, due to symmetry): C′=∫∫C(ϕ,θ)P(ϕ)sin(ϕ)dϕdθ where the first integral is evaluated from 0->pi/2, and second from 0->2pi P(ϕ) is the probability density "line". I normalized this line such that the area underneath the revolved surface about the X1 axis is 1. This method however is giving me results that don't make sense - the values are all too small. I feel like there is an error with either the fact that i am using a cartesian curve while integrating over a sphere, or with the normalized area of the probability density function. Anyone have any idea why this might be happening, or other ideas on how to approach the problem? Any help would be greatly appreciated. Thanks!