Finding the Interval for Theta in Parametric Representation of a Sphere

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SUMMARY

The discussion focuses on determining the interval for the angle theta in the parametric representation of a sphere defined by the equation x² + y² + z² = 16, specifically between the planes z = -2 and z = 2. The parametric equations provided are x = 4sin(φ)cos(θ), y = 4sin(φ)sin(θ), and z = 4cos(φ). The key insight is that theta represents the angle in the xy-plane, and its interval is restricted by the projection of the sphere's surface onto this plane, which is influenced by the limits set by the z-coordinate.

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eunhye732
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Find the parametric representation for the surface:
The part of the sphere x^2 + y^2 + z^2 = 16 that lies between the planes z = -2 and z = 2.

okay, i know that i have to use spherical coordinates which is
x = 4sin(phi)cos(theta)
y = 4sin(phi)sin(phi)
z = 4cos(phi)

i know how to find the interval for phi, but how do you find the interval for theta? this is probably a stupid question, but i don't get it.
thanks!
 
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eunhye732 said:
...
y = 4sin(phi)sin(phi)
...
That should be y=4sin(phi)sin(theta).
eunhye732 said:
...
i know how to find the interval for phi, but how do you find the interval for theta?
...
What is theta on the sphere ? If you set phi to some constant, what curve results on the sphere's surface ? What does this imply about the restriction on theta ?
 
Remember that [itex]\theta[/itex] measures the angle in a plane parallel to the xy plane. Imagine the sphere cut by such a plane for z between -2 and 2. What restrictions are there on [itex]\theta[/itex]?
 

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