The cylinder x^2 +y^2 = x divides the unit sphere
x^2+ y^2+ z^2=1 into two regions M and N
where M is inside the cylinder and N is outside.
Compute the ratio of the area of M to the area of N.

Question

The cylinder x^2 +y^2   = x divides the unit sphere  
x^2+ y^2+ z^2=1 into two regions  M and N
where M is  inside the cylinder and N is outside.
Compute the ratio of the area of M to the area of N.

anonymous · Answer

See the attached picture.

dumbcow · Answer

do you mean ratio of volumes? or surface area?
also is there a typo
x^2 + y^2 = x  is a circle
  --> (x- .5)^2 +y^2 = .25

dumbcow · Answer

also, if you can could go here and see if you can help answer this question http://openstudy.com/updates/4fa9f71ce4b059b524f5d875

anonymous · Answer

Surface areas

dumbcow · Answer

i get a ratio of:
$$\frac{M}{N} = \frac{2\pi -4}{2\pi +4}$$
i found area of inside by summing up arc lengths of segments of each circle of unit sphere that was inside cylinder
$$\rightarrow 2\int\limits_{0}^{1}r*\cos^{-1} (2r^{2}-1) dS$$
$$=2\int\limits_{0}^{1}\frac{r}{\sqrt{1-r^{2}}} \cos^{-1} (2r^{2}-1) dr$$
$$= 2\pi - 4$$

anonymous · Answer

x^2 + y^2 = x is a circle in the plane xy, but a cylinder in 3D where z is a free variable.