Sphere:

A spherical surface is produced when a semicircular arc ACB of diameter AB = 2 a revolves around the axis AB.

The radius of semicircle is a as illustrated in Figure.

(i) As finding out in Example, the centroid of the semicircular arc is at GL

where, x¯ =  2 a/ π  from central diameter AB. Length (L) of semicircle

ACB = π a.

∴  {Surface of sphere generated in one revolution }=  L × x¯ × 2 π

      = π a ×  (2 a / π )× 2 π

       = 4 π a ²

 (ii) Volume of sphere : The area A of the plane figure ACBA  =  (π a ² /2)  . The distance of the centroid G_A of this area is (4a/ 3 π) from AB,

                     Volume of the solid Sphere =  A × x¯ × 2 π

                                                                     =  (π a²/2) ×  (4 a/3 π )× 2 π =  4 π a³ /3