Average of a factor over $\phi$ and $\lambda$

Question

I am doing a project on map projections and want to find the average of the area inflation/deflation factor over the whole map. Of course I want a factor 1/100 to be just as prominent to the average as a factor 100, hence integration over $\phi$ ($-0.5\pi$ to $0.5\pi$) and $\lambda$ ($-\pi$ to $\pi$) of the area inflation factor $s(\phi, \lambda)$ does not feel right. How do I obtain a correct average of this? Do I use the log of the area inflation?