Surface area of a sphere-derivation

[muppet]

[SOLVED] Surface area of a sphere-derivation

This isn't really a homework question, it just would've been handy to be able to do for an electromagnetism problem last year, and has been bugging me since!
Is it possible to derive the surface area of a sphere by double integration?
At the time I tried diving the surface into many infinitesmal regions that could be considered approximately plane rectangles. Each of these regions had sides of length rd(phi) and rd(theta), where phi and theta are the polar and azimuthal angles. The area of these regions was therefore [tex]r^{2}d\theta d\phi[/tex]
Computing the integral over the limits (0, 2[tex]\pi[/tex]),(0,[tex]\pi[/tex])
you're out by a factor of 2/[tex]\pi[/tex].
Any suggestions?

 

[cristo]

Your surface area differential is incorrect. In spherical polars, it is not [itex]r^2d\theta d\phi[/itex], but is in fact [itex]r^2\sin\theta d\theta d\phi[/itex].

See this for a diagram: http://hyperphysics.phy-astr.gsu.edu/hbase/sphc.html

 

[muppet]

I figured it would be a correction for the tapering with height, but couldn't work out how- cheers

 

[cristo]

You're welcome.