This solution may seem trivial, since it's already been done, but for those interested, this page shows the calculus for deriving the surface area of a sphere.
> This solution may seem trivial, since it's already been done, but for > those interested, this page shows the calculus for deriving the surface > area of a sphere.
Suppose a sphere with center located at the origin is sliced by a plane parallel with the xz axis. This forms a circle. The xy plane slices the circle at a point in Quadrant I. The angle between this point, the origin and the x axis is angle w.
Angle w = (Arclength A)/(radius r)
w = A/r
wr = A differentiating,
rdw + wdr = dA The radius doesn't change, so dr=0
rdw = dA
C is the circumference of the parallels at each increment of dA. Each parallel has a radius of r cos w. A band of surface area dS of Circumference C is,
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