I need to compute the area of a $3D$ sphere centered on $0;0;0$ and the book I'm following says:
"If a curve $y = f (x)$ from $y = a$ to $y = b$ is revolved around the $x$ axis, the surface area of the resulting swept surface is"
What is this formula? I've never seen it before. I'd like to get a simple explanation on this or at least a simple proof why this should work
It's proved on the wikipedia page for surface of revolution (not solid of revolution). That page also covers the specific case of a sphere.