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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?

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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 $t=1$.

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  • $\begingroup$ Thank you. Would you be able to show me on the same example, but bounded by some other function like $y = x^2$ (or anything, really), so I can get a better grasp, please? $\endgroup$ – weno Apr 10 at 1:04
  • $\begingroup$ Oh, so bounded by $y = x^2$ I would need to solve something like this? $y(t) = (x(t))^2$ $\endgroup$ – weno Apr 10 at 1:05
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    $\begingroup$ Let's say your third bound is $y = x^2$. Then you would need to solve $(t^4-1)^2 = t^4-t^2$. $\endgroup$ – D.B. Apr 10 at 1:05

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