Derivation of Formula for Total Surface Area of the Sphere by Integration
The total surface area of the sphere is four times the area of great circle. To know more about great circle, see properties of a sphere. Given the radius r of the sphere, the total surface area is
From the figure, the area of the strip is
Where ds is the length of differential arc which is given by
See Length of Arc in Integral Calculus (link not yet active) for more information about ds.
The total area of the sphere is equal to twice the sum of the differential area dA from 0 to r.
From the figure,
x = r sin θ
dx = r cos θ dθ
When x = 0, θ = 0
When x = r, θ = π/2