In[1]:= AbsoluteTiming[
 h = 100.0;
 v = 20.0;
 g = 9.81;
 vol = Integrate[2 Pi x (h + v^2/(2 g) - g/2*x^2/v^2),
   {x, 0, (v Sqrt[2 g h + v^2])/g}];
 {IntegerPart[vol], N[FractionalPart[vol]]}]

Out[1]= {0.00168545, {1856532, 0.845528}}