Parametrization of intersection of two curves

Question

How to determine a parametrization of the border of the $R$ region in a clockwise direction, where $R$ is defined by

$y\geq \left | x \right |+2$ and $y\leq 4-x^{2}$

I try to solve the problem, but can not

Could someone please help? Thanks.

Answer 1

You may just split it into parts.

For the whole inferior part of the region, a parametrization should take the form $(x, |x|+2)$. As $x$ varies from $1$ to $-1$, we may choose $x=1-|t|$ for $0\le t<2$. Thus, $(x, |x|+2) = (1-|t|, |1-|t||+2)$. Another option could be splitting the inferior part into two subparts: one for $x>0$ and the other one, for $x<0$.

The superior part is more intuitive since $x$ grows as $t$ does. Letting $x=t-3$, the form $(x,4-x^2)$ turns into $(t-3,-t^2+6t-5)$ for $2 \le t \le 4$.

Therefore, a parametrization is $(1-|t|, |1-|t||+2)$ for $0\le t<2$ and $(t-3,-t^2+6t-5)$ for $2 \le t < 4$.