a  = 1.0;

delta = Pi/4.0;

h =  3/8 a Cos[Pi/2 + delta];     (* h is negative *)

Print[" h is ",h];

(*
ans = NIntegrate[p^2 Sin[f],

{f,ArcCos[h/a], Pi+delta},

{t,0,Pi},

{p, h Sec[f],a}] 

*)

theta[f_] := ArcSin[Cot[f] Tan[delta]];

ParametricPlot3D[{p Sin[f] Cos[theta[f]], p Sin[f] Sin[theta[f]], p Cos[f]},{f,delta,Pi+delta},
{p,0,1},PlotPoints->30];