Parametric Equation of a sphere

[anubis01]

Homework Statement

Find parametric equations for the part of sphere x²+y²+z²=9 that lies between the planes y=1 and y=2.

Homework Equations

The Attempt at a Solution

Okay knowing that the p=3 I wrote the parametric equations for a sphere as
x=3sin$$\phi$$cos$$\theta$$ y=3sin$$\phi$$sin$$\theta$$
z=3cos$$\phi$$ now the phi bound is 0<$$\phi$$<$$\pi$$
but I'm not sure what to write for the $$\theta$$ bound.

 

[Dick]

It would be easier if they said between z=2 and z=1, right? Then you would just have to restrict phi. There's no reason why you can't interchange say, y and z in your parametrization.

 

[anubis01]

Oh okay so I can say z=3cos$$\phi$$
and from there plug in the values to find the $$\phi$$ bound to be cos(1/3)^-1<$$\phi$$<cos(2/3)^-1 and the $$\theta$$ bound should just be 0< $$\theta$$<2$$\pi$$

 

[Dick]

Oh okay so I can say z=3cos$$\phi$$
and from there plug in the values to find the $$\phi$$ bound to be cos(1/3)^-1<$$\phi$$<cos(2/3)^-1 and the $$\theta$$ bound should just be 0< $$\theta$$<2$$\pi$$

Right. But they did say y=1 and y=2. You'll also have to tweak your parametrization so y=3*cos(theta) instead of z=3*cos(theta).

 

[anubis01]

oh so then phi bound should be 0<$$\phi$$<$$\pi$$ and since y=3cos$$\theta$$ the bounds for $$\theta$$ are
cos(1/3)^-1<$$\theta$$<cos(2/3)^-1

 

[Dick]

oh so then phi bound should be 0<$$\phi$$<$$\pi$$ and since y=3cos$$\theta$$ the bounds for $$\theta$$ are
cos(1/3)^-1<$$\theta$$<cos(2/3)^-1

Sorry, sorry! I meant to say your parametrization has z=3*cos(phi) and you want to change it so that y=3*cos(phi). Leave theta as it is.

 

[anubis01]

alright i think i got it now. So bound for theta 0<$$\theta$$<2$$\pi$$
y=3cos$$\phi$$ the bounds for phi are then
cos(1/3)^-1<$$\phi$$<cos(2/3)^-1

and the parametric equation for x=3sin$$\phi$$cos$$\theta$$
and z=3sin$$\phi$$sin$$\theta$$

 

[Dick]

alright i think i got it now. So bound for theta 0<$$\theta$$<2$$\pi$$
y=3cos$$\phi$$ the bounds for phi are then
cos(1/3)^-1<$$\phi$$<cos(2/3)^-1

and the parametric equation for x=3sin$$\phi$$cos$$\theta$$
and z=3sin$$\phi$$sin$$\theta$$

That looks good to me.

 

[anubis01]

alright thank you for all the help, its much appreciated.