I am familiar with how to find an area bounded by the parametric curve if I am given some $t \in <0,1>$ or such interval.
I can also find the area if I am told that the parametric curve is bounded by one (or combination) of those constant functions:
- $y = A \quad A\in\mathbb{R} \quad\text{(e.g. }y=\sqrt{2}\text{)}$
- $x = B \quad B \in \mathbb{R} \quad\text{(e.g. }x=1\text{)}$
Problem is: what if I have to calculate an area bounded by some parametric curve and some other non-constant function like $y=x$?
Say, for example:
an area bounded by $\Bigg(x(t)=t^4-1,y(t) = t^4-t^2\Bigg)$ and $y=x$? How should I find the integral's boundaries/limits?
You can find the time limits by setting $x(t) = y(t)$. $$t^4-1 = t^4-t^2 \implies t = \pm 1.$$ So, the bounded area lies between $t=-1$ and $t=1$.