Surface Area of a Sphere

Date: 04/10/98 at 14:01:19
From: Jichan Chong
Subject: How to prove the equation of surface area of a sphere

Hi! 

This is Jichan Chong. I am a junior student at HKIS (Hong Kong 
International School) high school, and I am taking AP-CALC now. 

One day I was proving the equation of volume of a sphere, which is 
4/3*pi*r^3. Well, I used an integral - actually, the revolutionary 
disc method - of the equation of a circle, x^2 + y^2= r^2, and I 
proved it, and then, suddenly I thought of proving the equation of the
surface area of a sphere. 

Firstly I simply checked it. I differentiated the equation 4/3*pi*r^3, 
and I got 4*pi*r^2, which is correct. Then I tried to prove it, and 
that was the start of my trouble. I tried so many methods to prove it, 
but I couldn't. 

My basic thought on this problem is that I get the surface area of a 
sphere when I integrate the diameter of radius r, from 0 to r. I spent 
plenty of time to solve this, but I couldn't get it. Would you please 
help me? My friends said that it's a useless thing to do since there's 
no possibility that such a question will be on the AP exam, but anyway 
I just want to know how to solve this problem.

Thank you.

Date: 04/10/98 at 17:47:58
From: Doctor Anthony
Subject: Re: How to prove the equation of surface area of a sphere

If you imagine the volume of a sphere made up of an infinite number of 
thin, hollow shells each of surface area A(r) where A(r) is the area 
of the shell when the radius is r, then we could say

    Volume  = INT[A(r)dr]  

since the element of volume is:

    surface area of shell * thickness of shell dr

So:

    (4/3)pi*r^3 = INT[A(r)dr]   

Differentiating both sides:

   (4*pi*r^2)dr = A(r)dr 

       4*pi*r^2 = A(r)

So the surface area of a sphere of radius r is given by 4*pi*r^2.

-Doctor Anthony,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/