Final answer:
The surface area of a sphere with a given radius a>0 can be found using surface integral methods by integrating the surface area element over the sphere's angles, leading to the well-known formula 4πa².
Step-by-step explanation:
To find the surface area of a sphere using a surface integral, recall the basic formula for the surface area of a sphere, which is 4πr². In this case, the radius is given by a. When applying the surface integral method, we are essentially summing up infinitesimally small elements of the surface area across the entire sphere.
The surface area element in spherical coordinates is given by dA = a² sin(θ)dθdφ, where θ is the polar angle and φ is the azimuthal angle. The integration then becomes a double integral over these two angles, and we integrate from 0 to π for θ and from 0 to 2π for φ. The complete integral to find the surface area is:
∫∫ a² sin(θ) dθ dφ = 4πa².