How to derive the volume of a sphere??

Asked by Viri Devarapalli | 20th Feb, 2014, 06:48: AM

Expert Answer:

begin mathsize 16px style Derivation space for space Volume space of space the space Sphere space
Figure space for space the space Derivation space of space Formula space of space Sphere space
by space IntegrationThe space differential space element space shown space in space the space figure space
is space cylindrical space with space radius space straight x space and space altitude space dy. space
The space volume space of space cylindrical space element space is... space
dV equals πx squared dy
The space sum space of space the space cylindrical space elements space from space 0 space to space straight r space is space straight a space hemisphere comma space
twice space the space hemisphere space will space give space the space volume space of space the space sphere. space
Thus comma space straight V equals 2 straight pi integral subscript 0 superscript straight r straight x squared dy space
From space the space equation space of space the space circle space straight x squared space plus space straight y squared space equals space straight r squared semicolon space straight x squared space equals space straight r squared space minus space straight y squared. space space
straight V equals 2 straight pi integral subscript 0 superscript straight r left parenthesis straight r squared minus straight y squared right parenthesis dy
space straight V equals 2 straight pi open square brackets straight r squared straight y minus straight y cubed over 3 close square brackets subscript 0 superscript straight r
space straight V equals 2 straight pi open parentheses open square brackets straight r cubed minus straight r cubed over 3 close square brackets minus left parenthesis 0 minus 033 right parenthesis close parentheses
space straight V equals 2 straight pi open square brackets fraction numerator 2 straight r cubed over denominator 3 end fraction close square brackets
straight V equals fraction numerator 4 πr cubed over denominator 3 end fraction end style

Answered by Arun | 7th Jan, 2018, 10:35: AM

Queries asked on Sunday & after 7pm from Monday to Saturday will be answered after 12pm the next working day.