Hi. Suppose that we use them to solve this kind of problem: find triple integral of dxdydz/(x^2+y^2+z^2) in region D that contains all points (x,y,z) such that: {x^2+y^2+z^2=1, x^2+y^2=z^2, |y|=x}.
I use spherical coordinates of the form
x=rcosφcosθ,
y=rsinφcosθ,
z=rsinθ,
After substituting
r∈(-∞,-1] and [1,+∞),
φ ∈ [-π/4+πk, π/4+πk],
θ ∈ [-π/4+2πk, π/4+2πk],
I hope it's correct. Can you tell me exact ranges I should take to finish counting integral?
Note: according to wiki: r ∈ [0, ∞), φ ∈ [0, 2π), θ ∈ [0, π] (I think this case there it is [-π/2, π/2])

Question

Hi. Suppose that we use them to solve this kind of problem: find triple integral of dxdydz/(x^2+y^2+z^2) in region D that contains all points (x,y,z) such that: {x^2+y^2+z^2>=1, x^2+y^2>=z^2, |y|>=x}.
I use spherical coordinates of the form 
x=rcosφcosθ,
y=rsinφcosθ,
z=rsinθ, 
After substituting
r∈(-∞,-1] and [1,+∞),
φ ∈ [-π/4+πk, π/4+πk],
θ ∈ [-π/4+2πk, π/4+2πk],
I hope it's correct. Can you tell me exact ranges I should take to finish counting integral? 
Note: according to wiki: r ∈ [0, ∞), φ ∈ [0, 2π), θ ∈ [0, π] (I think this case there it is [-π/2, π/2])