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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 
x-axis. 

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

Enjoy!

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


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! http://mathforum.org/dr.math/   

For more about sphere formulas, see the Dr. Math FAQ:

  http://mathforum.org/dr.math/faq/formulas/faq.sphere.html   
    
Associated Topics:
High School Calculus
High School Geometry
High School Higher-Dimensional Geometry

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