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Surface area of a sphere

  • Thread starter springo
  • Start date
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1. Homework Statement
Calculate the area for 3D sphere.

2. Homework 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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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.
 
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