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springo
#1
Mar17-09, 03:39 PM
P: 126
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:
[tex]A=2\pi\int_{a}^{b}f(x)\sqrt{1+ f'(x)^2}\:\mathrm{d}x[/tex]

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 [itex]2\cdot\pi\cdot r[/itex], with [itex]r=\sqrt{R^2-x^2}[/itex].
We could then integrate:
[tex]\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[/tex]
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)
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