The problem is to calculate the volume of the region contained within a sphere and outside a cone in spherical coordinates.
I am having difficulty converting the equation of the cone into spherical coordinates. Judging by the graph I was able to deduce from the formulas (working in rectangular coordinates), I believe I will need the cone equation as the inside ρ limit.
The Attempt at a Solution
Converting the sphere into spherical coordinates:
ρ=4 a sphere with radius 4 centered at the origin, which is consistent with my graph.
I recognize that the cone is downward opening and peaks at z=0. My attempt to convert the equation was as follows:
I can't figure out how to further simplify this formula. All the examples of cones in spherical coordinates I came across were peaked at the origin and simplified nicely to [itex]\phi[/itex]=(some arbitrary angle), but I couldn't find any that were more complicated.
This is my first post so please let me know if I've done anything wrong, and thanks in advance!