A sphere is a geometric figure obtained by the rotation of a circle X 2+Y 2 = R 2 around the X axis.

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)