Area of Ellipse/Proof 1

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:
 * $y = \pm b \sqrt {1 - \dfrac {x^2} {a^2} }$

From the geometric interpretation of the definite integral:

Let $x = a \sin \theta$ (note that we can do this because $-a \le x \le a$).

Thus:
 * $\theta = \arcsin \paren {\dfrac x a}$

and:
 * $\d x = a \cos \theta \rd \theta$

Then: