Surface area of a sphere

[springo]

1. The problem statement, all variables and given/known data
Calculate the area for 3D sphere.

2. Relevant equations
I know there's this formula for surface of revolution:
A=2\pi\int_{a}^{b}f(x)\sqrt{1+ f'(x)^2}\:\mathrm{d}x

3. The attempt at a solution
I thought of dividing the the sphere into slices, each of which contains a ring.
The length of each ring is 2\cdot\pi\cdot r, with r=\sqrt{R^2-x^2}.
We could then integrate:
\int_{-R}^{R}2\pi\sqrt{R^2-x^2}\:\mathrm{d}x=4\pi\int_{0}^{R}\sqrt{R^2-x^2}\:\mathrm{d}x=\pi R^2
But this is not correct so there must be something wrong...

PS: Just out of curiosity, is there any way to prove the formula for the surface are of an n-sphere using calculus? (the one with Γ)

[Feldoh]

The integral which you computed is (obviously) for area based on your answer. This is because you're taking a whole bunch of rings with an infinitely small width and summing them up from 0 to R. Geometrically think of it as taking a ring and fitting successively smaller rings inside of it until the point at which all the rings together resemble a single solid. That is why you are computing the area instead of surface area.

http://en.wikipedia.org/wiki/Hypersphere see the portion on volume.