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## Homework Statement

The problem is to calculate the volume of the region contained within a sphere and outside a cone in spherical coordinates.

Sphere: x

^{2}+y

^{2}+z

^{2}=16

Cone: z=4-√(x

^{2}+y

^{2})

## Homework Equations

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:

x

^{2}+y

^{2}+z

^{2}=16

ρ

^{2}=16

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:

ρcos[itex]\phi[/itex]=4-√(ρ

^{2}sin

^{2}[itex]\phi[/itex]cos

^{2}θ+ρ

^{2}sin

^{2}[itex]\phi[/itex]sin

^{2}θ)

ρcos[itex]\phi[/itex]=4-ρsin[itex]\phi[/itex]

ρ(cos[itex]\phi[/itex]+sin[itex]\phi[/itex])=4

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!