I had trouble understanding this formula, until I realized this but this just an example of the Mean Value theorem extended to 2 dimensions. Here we are saying that that the integral of a function $$L(x,\omega)$$ over the domain $[A, \Omega]$ is just the the average value of the function $\bar{L}$ multiplied by the area/surface patch of the domain $T$.

I had trouble understanding this formula, until I realized this but this just an example of the Mean Value theorem extended to 2 dimensions. Here we are saying that that the integral of a function $$L(x,\omega)$$ over the domain $[A, \Omega]$ is just the the average value of the function $\bar{L}$ multiplied by the area/surface patch of the domain $T$.