Describe using spherical coordinates the solid E in the first octant that lies above the half-cone z=√(x2+y2) but inside x2+y2+z2=1. Your final answer must be written in set-builder notation.
ρ = x2+y2+z2
x = ρsinφcosθ
y = ρsinφsinθ
z = ρcosφ
The Attempt at a Solution
Since we are in the first octant, θ will go from [0,π/2].
However the problem comes with describing ρ and φ,
Since we are in the first octant I believe that φ will be the same as θ, however for ρ,
I substituted in the relevant equations into both equations that were given.
ρ= ±1 <-- Unit sphere
ρ=√(2ρ2sin2φ)/cosφ <-- Half cone
Would I be able to use the positive bound of the unit sphere as the upper limit and the bound gotten from the cone as the lower limit?