Area of Ellipse/Proof 2

Proof
Let $K$ be an ellipse aligned in a cartesian coordinate plane in reduced form.

Then from Equation of Ellipse in Reduced Form:
 * $\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$

Thus:
 * $y = \pm \dfrac b a \sqrt{a^2 - x^2}$

Consider a circle of radius $a$ whose center is at the origin.

From Equation of Circle: Cartesian: Corollary 2, its equation is given by:
 * $x^2 + y^2 = a^2$

and so:
 * $y = \pm \sqrt{a^2 - x^2}$

The formulas show that each ordinate of the ellipse is $\dfrac b a$ the ordinate of the circle.

Since the same thing is true of the vertical chords:

Historical Note
This was the proof given by the method of.

It is essentially the same method as, except that while used chords to divide the area up,  used thin rectangles.