Suppose I have a thing such as an ellipse:
$$\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=1$$
now we can define it so that $\frac{x}{a}=cos(\theta)$ and $\frac{y}{b}=sin(\theta)$. I know the perimeter formula
$$\mu(S)=\int\sqrt{1+\left(f'(x)\right)^{2}} dx.$$
It is easy to paramerize the ellipse but how can I parametrize the perimeter formula so that I can easily calculate the perimeter?
I find that I am doing things the hard way like this:
$$y=\pm b \sqrt{1-\left(\frac{x}{a}\right)^{2}}$$
now if I plug in the y into the formula of perimeter, it is messy. Can I do it elegantly with parametric form somehow?