Area of Ellipse

Theorem
Let $K$ be an ellipse whose major axis is of length $2 a$ and whose minor axis is of length $2 b$.

The area $\mathcal A$ of $K$ is given by:
 * $\mathcal A = \pi a b$

Proof
Let $K$ be aligned in a cartesian coordinate plane such that:


 * The major axis of $K$ is aligned with the X-axis
 * The minor axis of $K$ is aligned with the Y-axis.

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

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:
 * $\theta = \arcsin \left({\dfrac x a}\right)$

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

Then: