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

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
$$\cal{A}=\int_0^{\pi} \int_0^1 1\cdot {r\over n}\;dr \;dt=\frac{\pi}{2n}$$
