This is a simple geometric derivation of the formula for the area of the surface of revolution:
If the surface area of a cone is expressed as:
equation sequence cap s sub c equals integral over zero under two times pi one divided by two times cap l times cap r d phi equals pi times cap r times cap l
Then the area of the frustum is the difference between the whole cone and the cut-away part:
equation sequence cap s sub f equals cap s minus s equals pi times cap r times cap l minus pi times r times open cap l minus l close equals pi times cap r times cap l minus pi times cap r times cap l minus l divided by cap l times open cap l minus l close equals pi times cap r times open cap l minus left parenthesis cap l minus l times right parenthesis squared divided by cap l close equals pi times cap r times l times open one plus cap l minus l divided by cap l close equals pi times open cap r plus r close times l
So, in the infinitesimal case:
equation sequence delta times cap s equals pi times open sum with, 3 , summands y plus y plus delta times y close times delta times l equals two times pi times y times delta times l plus o of delta times y equals two times pi times y times Square root of one plus y super prime two times delta times x plus o of delta times x
Surface of revolution
This is a simple geometric derivation of the formula for the area of the surface of revolution:
If the surface area of a cone is expressed as:
Then the area of the frustum is the difference between the whole cone and the cut-away part:
So, in the infinitesimal case: