As an example of Divergence Theorem, our textbook mentions finding area of an ellipse, but it isn't clear how it was derived though.
Following is an excerpt from the textbook.
Suppose there is an ellipse with the following equation, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ Then we could parameterize it into $x=a\cos t, y=b\sin t \space (0\le t\le 2\pi)$ In order to find the area enclosed by the ellipse, we use the equation $area=\frac 12\int\mathbf r\cdot \mathbf n \space ds$ where $\mathbf r(x,y)=x\mathbf i+y\mathbf j$ $\mathbf n=(b\cos t,a\sin t)/v(t)\space (v(t)=\sqrt{b^2\cos^2t+a^2\sin^2t})$ Hence, the area of the ellipse is $\begin{align} area & = \frac 12\int\mathbf r\cdot \mathbf n \space ds \\ & = \frac 12 \int_0^{2\pi} (a\cos t, b\sin t)\cdot(b\cos t, a\sin t)dt \\ & = ab\pi \end{align}$
What I'm not following is that how we could just ignore $v(t)$ when substituting $\mathbf n$? Or have I missed something along the way?
EDIT: I've actually tried throwing the entire equation (including $v(t)$) into wolframalpha, but, unfortunately, the engine couldn't return any meaning results (calculation timeout).