Re: [Phys-l] elliptical thinking

[John Denker <jsd@av8n.com>]

On 07/21/2008 12:27 PM, ludwik kowalski wrote:
> A have now read John's piece (clicking on his first link). Would a  
> math teacher call this a proof or an observation? Yes, the upper limit  
> for S is 4*a*b, while the lower is 2*a*b.  By why is 2 or 4 must be  
> replaced by pi? How do we know that what is true for a circle must  
> also be true for an ellipse?

  Note: By way of background, I assume everybody knows the formula for 
  the area of a circle,and the area of a square.

My point is that you can parlay that into the formula for the area of 
an ellipse ... by a scaling argument.

  Note:  If you want an absolutely rigorous mathematical proof of this 
  result, you can express the area in terms of an integral, and then 
  do a change of variable. The resulting factor of b/a then pops out 
  in front of the integral, since an integral is a linear operator.
  That means you don’t even need to evaluate the integral in order 
  to obtain the scaling result.

  The intuitive explanation for why this works is even simpler.
    http://www.av8n.com/physics/scaling.htm#sec-ellipse