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## Homework Statement

Derive the formula for surface area of a sphere using

**integration of circles**

## Homework Equations

Need to get : S = 4πr

^{2}

## 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

2πr

^{2}∫ (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.

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