(* spherical results : KD = 0, KQuad = 0, Kmonopole = 0.078*)

fint[n_, m_] := Integrate[x^n (1 - x^2)^m, {x, 0, 1}]
gint[n_, m_] := Integrate[x^n (1 - x^2)^m (1 - 2 x^2)^2, {x, 0, 1}]
KD = (6int[2, 1]/fint[2, 3/2] - 4fint[4, 0]/fint[4, 1/2])/5
KQuad = (6 fint[4, 1]/fint[4, 3/2] - 4 fint[6, 0]/fint[6, 1/2])/5
Kmon = (6fint[4, 1]/fint[4, 3/2] - 4 gint[2, 0]/gint[2, 1/2])/5
Kcompressional =
  6fint[4, 1]/fint[4, 3/2]/5 -
    4 (c^2 fint[2, 0] + 2 c fint[4, 0] + fint[6, 0])/(c^2 fint[2, 1/2] +

              2 c fint[4, 1/2] + fint[6, 1/2])/5
Kmonopole = Kcompressional /. {fint -> int, c -> -1/2}

(* axial trap results:  radial, axial modes *)
s = Sin[y];
c = Cos[y];
h1 = x^2*s^2 + 2/5*x^2*c^2 - 2/5
h2 = 1 - 6*x^2*c^2
H1 = Integrate[h1^2*s/2, {y, 0, \[Pi]}]
H2 = Integrate[h2^2*s/2, {y, 0, \[Pi]}]

A1 = 6fint[4, 1]/fint[4, 3/2]/5 -
    4*Integrate[
          H1*x^2, {x, 0, 1}]/(5*Integrate[H1*x^2*(1 - x^2)^(1/2), {x, 0,
1}])
A2 = 6 fint[4, 1]/fint[4, 3/2]/5 -
    4*Integrate[
          H2*x^2, {x, 0, 1}]/(5*Integrate[H2*x^2*(1 - x^2)^(1/2), {x, 0,
1}])