A simple question regarding spheres and spherical shells

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Moved my question to Intro Physics. Mods feel free to delete.
 
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I found the answer online. Apparently

\sigma=\rho dr'.

This doesn't really make sense to me. Can anyone clear up why this is so?
 

Thread 'Modeling a graph that shows age in relation to depth of an ice sample'

To solve this, I first used the units to work out that a= m* a/m, i.e. t=z/λ. This would allow you to determine the time duration within an interval section by section and then add this to the previous ones to obtain the age of the respective layer. However, this would require a constant thickness per year for each interval. However, since this is most likely not the case, my next consideration was that the age must be the integral of a 1/λ(z) function, which I cannot model.
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