Question

The Volume of a Sphere.

Calculate the volume of the sphere of radius 1, using spherical shells by imagining the sphere as being constructed from an infinitude of spherical shells emanating from the origin and integrating the expression for the volume of each of the shells.

Answer 1

Imagine a sphere of radius 1. Now, let us consider a shell of thickness 'dr' at a radius 'r' from the center.

Then, the volume of the considered is given by,

Volume of shell = Volume of sphere of radius 'r + dr' - Volume of the sphere with radius 'r'

= 4 π (r+dr)³ / 3 - 4 π (r)³ / 3

= 4 π ( r³ + dr³ + 3 r² dr + 3 r dr² - r³) / 3

As, dr is very small when compared to r, higher powers of dr can be neglected. By which, we have dr² = dr³ = 0.

=> Volume of spherical shell = 4 π ( 3 r² dr ) / 3

= 4 π r² dr

=> The total volume of the sphere is given by the sum of all such spherical shells that are considered.

=> Volume of sphere

  

= [ 4 π r³ / 3 ]₀¹

= 4 π (1³ - 0³) / 3

= 4 π / 3.

Hence, the volume of the sphere of radius 1 is obtained as "4π/3".