With the following code, I can only get the results when the radius of the ball and the half apex angle of the cone are specific values.
Clear["Global`*"];
v1 = ImplicitRegion[x^2 + y^2 + (z - 2)^2 <= 2^2, {x, y, z}]
v2 = ImplicitRegion[z >= Cot[Pi/3]*(\!\(TraditionalForm\`
\*SqrtBox[\(
\*SuperscriptBox[\(x\), \(2\)] +
\*SuperscriptBox[\(y\), \(2\)]\)]\)), {x, y, z}]
v = RegionIntersection[v1, v2];
Assuming[a > 0 && Pi/2 > \!\(TraditionalForm\`\[Alpha]\) > 0,
Volume[v]]
(*10Pi*)
But very long time by:
Clear["Global`*"];
v1 = Ball[{0, 0, 2}, 2];
v2 = Cone[{{0, 0, 0}, {0, 0, 2*2}}, 2*2*Tan[Pi/3]];
v = RegionIntersection[v1, v2];
Volume[v]
(10Pi)
However, I can't get the result when the radius of the ball and the half apex angle of the cone are variable with the following code:
Clear["Global`*"];
v1 = ImplicitRegion[x^2 + y^2 + (z - a)^2 <= a^2, {x, y, z}]
v2 = ImplicitRegion[z >= Cot[\[Alpha]]*(\!\(TraditionalForm\`
\*SqrtBox[\(
\*SuperscriptBox[\(x\), \(2\)] +
\*SuperscriptBox[\(y\), \(2\)]\)]\)), {x, y, z}]
v = RegionIntersection[v1, v2];
Assuming[a > 0 && Pi/2 > \!\(TraditionalForm\`\[Alpha]\) > 0,
Volume[v]]
($Aborted)
How to get the result?
FindSequenceFunction
approach and you want something more automated? $\endgroup$