Derive the formula for surface area of a sphere using integration of circles
Need to get : S = 4πr2
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) plane 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.