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Real constant π

π is the area of the unit circle x2 + y2 = 1,

 


Explanation

The unit circle can be written as the equation

The area of the circle is twice the antiderivative between −1 < x < 1. From the table of integrals we use the formula

 

and substitute a = 1 and u = x, so

 


Example 1

It follows by a change of variables that the circle x2 + y2 = r2 of radius r has the area A = πr2. So for the unit circle, where r = 1 this becomes A = π.

 


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