# Calculating the volume of a paraboloidal region

I have an inequality that defines a paraboloid. I would like to calculate the volume of the intersection of this paraboloid and a cylinder of radius R.

ff[a_, d_, x_, y_, z_] :=
16 a^2 y^4 + a y^3 (-32 a^2 - 8 d^2 - 8 x^2 - 8 z^2) +
a y (32 a^2 d^2 + 8 d^4 - 8 a^2 x^2 + 2 d^2 x^2 - 10 x^4 -
8 a^2 z^2 + 2 d^2 z^2 - 20 x^2 z^2 - 10 z^4) +
y^2 (16 a^4 - 8 a^2 d^2 + d^4 + 32 a^2 x^2 - 2 d^2 x^2 + x^4 +
32 a^2 z^2 - 2 d^2 z^2 + 2 x^2 z^2 + z^4) - (16 a^4 d^2 +
8 a^2 d^4 + d^6 + 20 a^2 d^2 x^2 - 3 d^4 x^2 - a^2 x^4 +
3 d^2 x^4 - x^6 + 20 a^2 d^2 z^2 - 3 d^4 z^2 - 2 a^2 x^2 z^2 +
6 d^2 x^2 z^2 - 3 x^4 z^2 - a^2 z^4 + 3 d^2 z^4 - 3 x^2 z^4 - z^6);

Vol[R_, a_] :=
Integrate[
Boole[ff[a, d, x, y, z] <= 0 && x^2 + z^2 <= R^2],
{x, -R, R}, {y, -20, 20}, {z, -R, R}];


My problem is that Mathematica does not yield a result. Is there any way to do it?.

Clear["Global*"]

ff[a_, d_, x_, y_, z_] :=
16 a^2 y^4 + a y^3 (-32 a^2 - 8 d^2 - 8 x^2 - 8 z^2) +
a y (32 a^2 d^2 + 8 d^4 - 8 a^2 x^2 + 2 d^2 x^2 - 10 x^4 -
8 a^2 z^2 + 2 d^2 z^2 - 20 x^2 z^2 - 10 z^4) +
y^2 (16 a^4 - 8 a^2 d^2 + d^4 + 32 a^2 x^2 - 2 d^2 x^2 + x^4 +
32 a^2 z^2 - 2 d^2 z^2 + 2 x^2 z^2 + z^4) - (16 a^4 d^2 +
8 a^2 d^4 + d^6 + 20 a^2 d^2 x^2 - 3 d^4 x^2 - a^2 x^4 +
3 d^2 x^4 - x^6 + 20 a^2 d^2 z^2 - 3 d^4 z^2 - 2 a^2 x^2 z^2 +
6 d^2 x^2 z^2 - 3 x^4 z^2 - a^2 z^4 + 3 d^2 z^4 - 3 x^2 z^4 -
z^6);

Vol[R_?Positive, a_?NumericQ, d_?NumericQ] :=
NIntegrate[
Boole[ff[a, d, x, y, z] <= 0 && x^2 + z^2 <= R^2], {x, -R,
R}, {y, -20, 20}, {z, -R, R}];

Vol[10, 1, 1]

(* 1599.87 *)

reg[R_?Positive, a_?NumericQ, d_?NumericQ] :=
ImplicitRegion[
ff[a, d, x, y, z] <= 0 &&
x^2 + z^2 <= R^2 && -R <= x <= R && -20 <= y <= 20 && -R <= z <=
R, {x, y, z}]

Volume[reg[10, 1, 1]]

(* 1599.87 *)

• thanks, but Is it possible to find an analytical expression? – F.Mark Jan 7 at 2:17
• Without using numeric values, determining when ff[a, d, x, y, z] <= 0` would seem unlikely. – Bob Hanlon Jan 7 at 2:58
• thanks so mucho – F.Mark Jan 8 at 2:13
• @F.Mark perhaps you could either use this numerical procedure and plot some dependencies? Or you could try to nondimensionalize your function to use less variables? – CA Trevillian Jan 11 at 23:15