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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

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

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

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

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