When -R≤x≤R, the chord length perpendicular to the x axis y = √ (r 2-x 2).
Here, we take a infinitesimal with a base radius of r=y and a height of h=dx.
Then the volume element and surface element of the sphere are the volume and side area VII of the infinitesimal (cylinder with R = Y and H = DX) respectively.
dS=2πydx,dV=πy^2dx
∴s=∫(-r,r)2πydx=∫(-r,r)2π√(r^2-x^2)dx=4πr^2,
v=∫(-r,r)π(y^2)dx=∫(-r,r)π(r^2-x^2)dx=4π/3*(r^3)