Area of Circle/Proof 6

Proof
From Equation of Circle:
 * $x^2 + y^2 = r^2$

Let $A$ be the area of the circle whose equation is given by $x^2 + y^2 = r^2$.

We have that:
 * $y = \pm \sqrt {r^2 - x^2}$

For the upper half of the circle:
 * $y = +\sqrt {r^2 - x^2}$

Thus for the right hand half of the upper half of the circle:

Hence the result.