Initial definitions

The integrand
In[]:=
f=ρ/(1+ρ^2)^(5/2)
Out[]=
ρ
5/2
(1+
2
ρ
)
Avoid error messages associated with the upper limit in the numerically evaluated integral
In[]:=
g[lower_,upper_?NumberQ]:=NIntegrate[f,{ρ,lower,upper}]
List of radii
In[]:=
r={0}
Out[]=
{0}
List of sector counts
In[]:=
s={8,16,24,24,24,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,32,32,16}
Out[]=
{8,16,24,24,24,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,32,32,16}

Ring radii and thicknesses

In[]:=
Do[AppendTo[r,x/.FindRoot[g[r[[i]],x]==.001/(3/s[[i]]),{x,r[[i]]+.1}]],{i,Length[s]}]
In[]:=
r
Out[]=
{0,0.0732744,0.127777,0.182585,0.226003,0.263816,0.330484,0.390801,0.448067,0.504125,0.560247,0.617473,0.676782,0.739211,0.805962,0.87854,0.958952,1.05003,1.15606,1.28396,1.44608,1.66772,2.01358,2.41493,3.31945,4.89898}
In[]:=
t=Table[r[[i+1]]-r[[i]],{i,1,Length[s]}]
Out[]=
{0.0732744,0.0545026,0.0548075,0.0434184,0.0378135,0.0666672,0.0603169,0.0572662,0.0560581,0.0561221,0.0572261,0.0593093,0.0624287,0.0667509,0.0725782,0.0804115,0.0910824,0.106026,0.127898,0.162126,0.221638,0.345857,0.401353,0.904518,1.57953}