(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the surface area of the surface [tex] x^2 + y^2 + z^2 = a^2 [/tex], that is outside the two cylinders [tex] x^2+y^2=\pm ax[/tex].

[tex](a>0)[/tex]

2. Relevant equations

[tex]ds= \sqrt{1+(Z_x')^2+(Z_y')^2}dxdy[/tex]

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.

[tex]S = \iint_D \sqrt{\frac{a^2}{a^2-x^2-y^2}}dxdy[/tex] when D is a circle given by [tex] \left(x-\frac{a}{2}\right)^2 + y^2 \le \left(\frac{a}{2}\right)^2 [/tex]

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 [tex]0<\phi < \frac{\pi}{2} , -a\cos\phi<\sin^2\theta < a\cos\phi[/tex] Which ain't nice either.

ANY help would be appreciated, especially before my exam on Sunday

Thanks,

Jaron

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# Homework Help: Surface of a sphere intersecting two cylinders

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