How to calculate area of an ellipse based on its formula?

Question

How can I determine the area of a half-ellipse if all that is given is $y = \sqrt{1-n^2x^2}$? I have tried both geometry and calculus, but without convincing results…

Thank you

Answer 1

The area of the half-ellipse is given by the integral $\displaystyle\int_{-1/n}^{1/n} ydx=2\int_0^{1/n}\sqrt{1-n^2x^2}\ dx$

There is a formula for solving integrals of the form $\int\sqrt{a^2-x^2}\ dx$, but we will not use it directly. Instead, substitute $nx=\sin\theta\implies dx=\frac{\cos\theta}nd\theta$

$\implies A=\frac2n\int_0^{\pi/2}\cos^2\theta d\theta=\frac1n\int_0^{\pi/2}(\cos2\theta+1) d\theta=\frac\pi{2n}$

Can you now show similarly that the area enclosed by the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is equal to $\pi ab$?

Answer 2

With the use of generalized polar coordinates \begin{aligned}x&=ar\cos t\\ y&=br\sin t\end{aligned} where $a=\frac 1n,\; b=1,\; t \in [0,\pi]\; \text{and}\; r\in [0,1]$ in the given case. The Jacobian is ${r\over n}$ and the area
$$\cal{A}=\int_0^{\pi} \int_0^1 1\cdot {r\over n}\;dr \;dt=\frac{\pi}{2n}$$