I have actually a question around the area enclosed between the complying with parametric equations:

eginalign* x &= t^3 - 8t \ y &= 6t^2endalign*

I understand the area is the integral that the $y(t)$ time the derivative of $x(t)$. What ns don"t know is just how to find the boundaries of integration because that $t$.

You are watching: Find the area of the region enclosed by the parametric equation

Thank you!



by drawing a graph, e.g.


you deserve to see the the loop is about points where $x = 0, y e 0$, that is $ t^3 - 8t = 0, t = +/- sqrt8$, these space your limits, then together you said

$A = intlimits_-sqrt8^sqrt8 y(t) x"(t) dt = 1303.3...$



Use Green"s thm in between $t$ borders $pm 2 sqrt2$ the encloses a loop between the origin and $ y=48 $


The graph has symmetry in the $y$ axis. The graph intersects v the $y$ axis when $t=0$ and also $t=pm2sqrt2$

You therefore need to calculation $$A=2int_t=0^t=2sqrt2xfracdydtdt$$

Take the hopeful value that this.


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