circle to sphere integration

1. The problem statement, all variables and given/known data



Derive the formula for surface area of a sphere using integration of circles





2. Relevant equations



Need to get : S = 4πr²





3. The attempt at a solution



Consider a sphere of radius r centred on the origin of a 3D space. Let y be an axis thru the origin. The sphere can be sliced into a row of circles with the y axis at their centres. Consider one of these circles. The circumference c of the circle depends on it centre's distance from the origin such that

c = 2πr cos θ , where θ is the angle from the origin between the (x,z) plain to any point on the circle's circumference. r cosθ is the radius of the circle.



For -r to r, y= r sinθ

The Integration the circles circumferences along y from -r to r is

2πr cos θ ∫dy

Since y = r sinθ, then dy / dθ = r cosθ , so dy = r cos θ dθ

so the integration formula is

2πr² ∫ (cos θ)² dθ



I don't get the right formula from that integral.

There's something wrong with my setting up the integral. I can't see what though. Please give me a hint or two.

via Physics Forums RSS Feed http://www.physicsforums.com/showthread.php?t=707879&goto=newpost