Question

A solid lies above the cone z = x2 + y2 and below the sphere x2 + y2 + z2 = z. write a description of the solid in terms of inequalities involving spherical coordinates

Answer 1

Since solid is above the cone z >= √x^2 + y^2 or z^2 >= x^2 + y^2 or 2z^2 >= x^2+z^2 = p^2 or 2p^2 cos^2 φ >= p^2. Thus cos φ >= 1/√2 since the cone upwards. Thus 0 <= φ <= pi /4.

On the other hand in spherical coordinates the sphere here is pcos φ = p^2 so 0 <= cos φ since the solid lies between the sphere. Thus, making it the two inequalities.