Hi,
actually, this is function expressed in polar coordinates, and "f" only gives the radius (which can be negative).
Defining
f[Theta_, Phi_] :=
a*Cos[Theta]^3 + b*Cos[Theta] Sin[Theta]^2 -
c*(Cos[Phi] - 3 Cos[3 Phi]) Sin[Theta]^3
I find that
Integrate[Sin[Theta]*f[Theta, Phi], {Theta, 0, Pi}, {Phi, 0, 2 Pi}]
is null. But this is not the right way to measure the volume enclosed in the surface (3 lobes), which is certainly not zero: see
FullSimplify[Sin[Theta]*f[Theta, Phi], Trig -> True]
Print[SphericalPlot3D[% /. {a -> 1, b -> 1, c -> 1}, {Theta, 0,
Pi}, {Phi, 0, 2 Pi},
PlotStyle -> Directive[Cyan, Opacity[0.7], Specularity[White, 10]],
Mesh -> None, PlotPoints -> 50, AspectRatio -> Automatic,
ImageSize -> Large]]
So, this is a bit more complicated than merely integrating Sin[Theta]*f[Theta, Phi]...