Problem 5 
Repeat for Hard Sphere

Clear["Global`*"]

π/Degree *1.

f[θ_]=(1/((Sin[θ/2 Degree])^4));

f[θ_]=1;

LogPlot[f[θ] Sin[θ Degree],{θ,0,180}]

Plot[f[θ] Sin[θ Degree],{θ,0,180}]

tmp=Integrate[f[θ] Sin[θ Degree],{θ,0,180},{ϕ,0,2π}]

int[a_,b_]:=NIntegrate[f[θ] Sin[θ Degree],{θ,a,b},{ϕ,0,2π}]

total=int[0,180]

Table[xint[a,a+10],{a,00,170,10}]

tab=Table[((int[a,a+10])/total),{a,0,170,10}]

Plus @@ tab

labels=Table[a,{a,0,170,10}]

Length /@ {labels,tab}

BarChart[ 1000 tab,ScalingFunctions->"Log",ChartLabels->labels]

BarChart[1000 tab,ScalingFunctions->"Linear",ChartLabels->labels]