[tolo]

I'm not sure if I'm missing something, but I get a much simpler answer. Surface area of a spherical cap is 2 pi R h, with h = R - r. Divided by the total surface area (4 pi R^2), gives (R-r)/(2 R)

[svrocks]

2 pi R h gives you the surface area of the side of a cylinder with height h, since you are taking the circumference of a circle and multiplying it by h

[tolo]

And also the surface area of a spherical zone, as shown here: http://mathworld.wolfram.com/Zone.html

[svrocks]

After choosing a different way to find the angle, my answer matches yours. Kudos for having a much simpler answer.

[svrocks]

wow that's an interesting and counterintuitive result. I need to figure out why my integral doesn't line up with that