Spherical Coordinates for Sphere Between Z-Planes

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Homework Statement



Find a parametric representation for the part of the sphere x^2 + y^2 + z^2 = 64 that lies between the planes z = -4 and z = 4.

i have found
x = 8sin(ϕ)cos(θ)
y = 8sin(ϕ)sin(θ)
z= 8cos(ϕ)

0 ≤ θ ≤ 2π (n looking thing is pi)

now i need to find the bounds of ϕ

i know under perfect conditions they are 0 to π
but these are not perfect conditions and I am not sure how to find them exactly?
 
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nevermind i got it.
what you do is take your z component and set it equal to you plane so
8cos(ϕ) = 4 you get cos(ϕ)=.5 so you get ϕ= pi/3
then by symmetry you know your lower bound will be pi - (pi/3) which give you 2pi/3
these are the correct bounds :)
 

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