Describing a region using spherical coordinates

[Kaleem]

Homework Statement

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.

Homework Equations

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?

 

[haruspex]

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?

Yes.

x = ρsinφcosθ
y = ρsinφcosθ

One of those should be sin θ.

ρ=√(2ρ²sin²φ)/cosφ

Where does the 2 come from? Can you not simplify this?

 

[Kaleem]

Yes.

One of those should be sin θ.

Where does the 2 come from? Can you not simplify this?

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?

 

[haruspex]

would the lower bound for ρ be π/4 or would that just be the upper bound for φ?

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?

 

[Kaleem]

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?

You get z=√(x2+y2) ≤ ρ ≤1

 

[haruspex]

You get z=√(x2+y2) ≤ ρ

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?