Final answer:
The surface area of a sphere of radius r can be determined through double integration in spherical coordinates. The total surface area is found by integrating the differential surface element over the range of the coordinates, yielding the result 4πr^2.
Step-by-step explanation:
To find the surface area of a sphere using surface integration with double integrals from calculus, you would usually start with a parameterized representation of the sphere surface. This representation can be expressed in spherical coordinates (r, θ, φ) as r = r, θ = θ, φ = φ, with r constant and θ and φ varying over [0, 2π) and [0, π] respectively.
The differential surface element in spherical coordinates is given by ds = r^2 sin(φ) dφ dθ. Integrating this surface element over the entire surface of the sphere gives the double integral ∫∫ds = ∫ (from 0 to 2π) ∫ (from 0 to π) r^2 sin(φ) dφ dθ, which evaluates to 4πr^2. Thus, the surface area of a sphere of radius r is indeed 4πr^2.
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