Homework Help: Surface of a sphere intersecting two cylinders

1. Aug 15, 2008

jarondl

1. The problem statement, all variables and given/known data

Find the surface area of the surface $$x^2 + y^2 + z^2 = a^2$$, that is outside the two cylinders $$x^2+y^2=\pm ax$$.
$$(a>0)$$

2. Relevant equations

$$ds= \sqrt{1+(Z_x')^2+(Z_y')^2}dxdy$$

3. The attempt at a solution

The intersections are clearly four ellipses. I will try to find a area of one of these ellipses in order to multiply it by four and subtract that from the known area of a whole sphere.
I have tried two different attitudes to solve it. First, I used the equation mentioned above, to convert this integral to a double integral.
$$S = \iint_D \sqrt{\frac{a^2}{a^2-x^2-y^2}}dxdy$$ when D is a circle given by $$\left(x-\frac{a}{2}\right)^2 + y^2 \le \left(\frac{a}{2}\right)^2$$
Alas, I haven't succeeded in solving this one (not when staying cartesian, nor with polar co-ordinates).
The second idea was to use spheric coordinates in the first place, but then the limits for theta and phi turn out to be $$0<\phi < \frac{\pi}{2} , -a\cos\phi<\sin^2\theta < a\cos\phi$$ Which ain't nice either.
ANY help would be appreciated, especially before my exam on Sunday
Thanks,
Jaron

2. Aug 15, 2008

tiny-tim

Welcome to PF!

Hi Jaron! Welcome to PF!
Nooo … it's not even a plane figure … an ellipse is a plane figure, and any plane intersects a sphere in a circle, but these cylinders in ellipses.
hmm … clever isn't always the easiest.

have you tried the dumb approach, just using horizontal slices of thickness dy?

(and didn't Archimedes have a theorem about areas on spheres and cylinders?)

3. Aug 17, 2008

jarondl

Re: Welcome to PF!

Thanks tiny Tim!

Yep.. You were right.
About Archimedes, I only find his ideas over a cylinder of the same radius as the sphere.

Anyway, thanks for you help