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Volume of a Sphere

Date: 04/21/98 at 14:08:55
From: Ariel Fishman
Subject: Calc 2 proof

Can you help me prove the formula for the volume of a sphere? I have 
no clue as to where to start! Your help with previous questions has 
been greatly appreciated and very helpful. Thank you!

--Ariel Fishman

Date: 04/21/98 at 17:15:31
From: Doctor Jen
Subject: Re: Calc 2 proof

Think about a circle, radius a, centre (0,0). The equation of this 
circle is x^2 + y^2 = a^2, yes? If you've done circles, you'll know 
this - if not, use Pythagoras' theorem to convince yourself.

You get a sphere by taking this circle and rotating it about the 

Once you have your sphere, imagine it chopped up into lots and lots 
of slices, cuts going vertically. It's like when you slice a tomato. 
One of these slices, we say, has radius y, and width (delta x). 

The volume of this element is approximately equal to pi*y^2*(delta x) 
(the area of the disc, times its thickness. It's approximate, because 
it's not quite a regular cylinder, being a slice of a sphere - the 
radius at the top is a bit bigger than the radius at the bottom).

The volume of the sphere, then, is the sum of all these elements, 
added together between x = -a and x = a. (Please try doing a diagram 
to follow this through; it should make a lot more sense. Well, when 
you've sliced the tomato, you put the slices together to make the 
whole again - yes?)

Thus, V is approximately SIGMA (pi*y^2*(delta x)).

This is still approximate, because we assumed the element was a disc 
in order to calculate its volume, when in fact its edge was slightly 
curved, so it wasn't a true cylinder.

So we let delta x -> 0. This makes the element become closer to being 
a cylinder as its width becomes smaller.

Now we can say V = int. pi*y^2 dx between -a and a.

If you remember, y^2 = a^2 - x^2. So substitute this into the 
integral, to get   V = int. pi*(a^2-x^2) dx between -a and a. 

Okay from there? Integrate, apply the limits, and find that 
V = (4*pi*a^3)/3


-Doctor Jen,  The Math Forum
Check out our web site!   

Date: 04/21/98 at 17:43:46
From: Doctor Anthony
Subject: Re: Calc 2 proof

There are many different methods, all using integral calculus.

If you consider a circle x+2 + y^2 = a^2 and rotate this about the x 
axis you generate a sphere. Using the usual formula for a volume 
of revolution:

     V = INT(-a to a)[pi*y^2*dx]

       = INT[pi*(a^2-x^2)dx]

       = pi*[a^2x - x^3/3]  evaluated from -a to a

       = pi[a^3 - a^3/3 -(-a^3 + a^3/3)]

       = pi[2a^3/3 + 2a^3/3]

       = 4pi*a^3/3

-Doctor Anthony,  The Math Forum
Check out our web site!   

For more about sphere formulas, see the Dr. Math FAQ:   
Associated Topics:
High School Calculus
High School Geometry
High School Higher-Dimensional Geometry

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