Use our studies about surface integral to parametrize the surface of a sphere of radius R center at the origin.

Question

Use this parametrization to set up the integral to compute the surface area of a sphere of radius R and evaluate the integral.

Answer 1

Envision a circle with center at the origin of radius R.

Let a circumference of the sphere be a circle y distance above the xy plane. Let t be the angle above the xy plane formed by a radius line from the origin to the intersection of the circumference line.

The circumference becomes 2pi(r) where r = Rcost. Imagine the differential of surface area of the sphere to be 2pi(r)*Rdt =2piR²costdt = dS. The surface area of the top half of the sphere would then be the integral from 0 to pi/2 and the surface area of the entire sphere would be twice that value. Completing the integral and multiplying by 2 yields 4piR²