# Thread: Surface area problem (in fact, more of a parametrization problem...)

1. ## Surface area problem (in fact, more of a parametrization problem...)

Hi all,

The cylinder x^2 + y^2 = x divides the unit sphere S into two regions S1 and S2, where S1 is inside the cylinder and S2 outside. Find the ratio of the Areas A(S2)/A(S1).

I know the surface area formula, but I can't seem to find a parametrization of the said surfaces that works! The book gives the answer (Pi+2)/(Pi-2) but there is no procedure showing how to get there.

Julian

2. Originally Posted by aznmaven
Hi all,

The cylinder x^2 + y^2 = x divides the unit sphere S into two regions S1 and S2, where S1 is inside the cylinder and S2 outside. Find the ratio of the Areas A(S2)/A(S1).

I know the surface area formula, but I can't seem to find a parametrization of the said surfaces that works! The book gives the answer (Pi+2)/(Pi-2) but there is no procedure showing how to get there.

Julian

3. Thanks for the quick reply.

Aren't those two different problems though? One is finding volume and my problem is surface area of some funky surfaces...

I presume I'd have to use some double integral of the norm of the vector normal to the surface (easily obtained after a parametrization of the surfaces in question is achieved), or is there something I'm missing here?

4. Originally Posted by aznmaven

Aren't those two different problems though? One is finding volume and my problem is surface area of some funky surfaces...

I presume I'd have to use some double integral of the norm of the vector normal to the surface (easily obtained after a parametrization of the surfaces in question is achieved), or is there something I'm missing here?
Sorry, my mistake.

I'd parametrise the sphere as follows:

$x = 1 \sin \phi \cos \theta$

$y = 1 \sin \phi \sin \theta$

$z = 1 \cos \phi$

so that $dS = 1^2 \sin \phi \, d \theta \, d \phi$.

$0 \leq \theta \leq 2 \pi$ is the polar angle.

Your task is to get the integral terminals for $\phi$.

5. Wow I've spent most of the night thinking about how to set the bounds for this problem, to no avail. Another hint, at least?
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