Find the volume of the solid that lies within the sphere x^2+y^2+z^2=16, above the xy plane, and outside the cone z=2sqrt(x^2+y^2).

Answer 1

Let Θ be the angle between the z axis and the edge of the cone. The equations above can be manipulated to show that Θ = arctan(1/2)

The volume of a cone in terms of the slant height, r, and angle Θ is (2 π /3) r³ ( 1- cosΘ) . Here r = 4

The volume outside the cone in the upper hemisphere is thus

(2 π /3) r³ - (2 π /3) r³ ( 1- cosΘ) = ( 2 π /3) r³ cosΘ

Noting that cos (arctan(1/2) ) = 2/sqrt(5) and that r = 4, The answer is:128 π /(3 sqrt(5))