# Describing a region using spherical coordinates

1. Oct 12, 2016

### Kaleem

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
ρ = x2+y2+z2
x = ρsinφcosθ
y = ρsinφsinθ
z = ρcosφ

3. 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?

Last edited: Oct 12, 2016
2. Oct 12, 2016

### haruspex

Yes.
One of those should be sin θ.
Where does the 2 come from? Can you not simplify this?

3. Oct 12, 2016

### Kaleem

You're right, I made a mistake when i factored it out, if i simplify I get 1 = tanφ, which gives me φ= π/4. Which confuses me, would the lower bound for ρ be π/4 or would that just be the upper bound for φ?

4. Oct 12, 2016

### haruspex

how does tan φ < π/4 turn into a bound on ρ?
Pick a point in the region. Vary ρ up and down until you hit the boundaries of the region. Which boundaries do you hit?

5. Oct 12, 2016

### Kaleem

You get z=√(x2+y2) ≤ ρ ≤1

6. Oct 12, 2016

### haruspex

Although that is true, it does not serve as a limit on ρ in polar coordinates. The limits should not mention the Cartesian coordinates.
If the point (ρ, θ,φ) is in the region, is the point (ρ/2, θ, φ) also in the region?