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Calculus of Parametric Equations

Parametric equations can be quite handy, and we don't want to unravel them just to do Calculus. Some tricks can bend traditional derivative and integral methods to apply to parametric equations.

Level 3

         

If \(y\) is a function of \(x\) as defined by the parametric relation \(y=3\sin ^{ 2 }{ t }\) when \(x=\tan { t } \), then determine the value of \(\displaystyle\lim_{ x\rightarrow \infty }{ y }\) .

The parametric equation of a cycloid is given below.

\[\large x = a(t - \sin t) \\ \large y = a(1 - \cos t)\]

What is the area of the region bounded by the two arcs of the cycloid in the above figure?

What is the length of the arc of the curve \(x^{\frac{2}{3}} + y^{\frac{2}{3}}=4\)?

The graph above satisfy the equation \(x^4 + y^3 = x^2 y \).

The area enclosed by the 2 cute adorable little fine loops is equals to \( \frac {a}{b} \) for coprime positive integers \(a\) and \(b\). What is the value of \(a+b\)?

The location of a dot \(P\) at a given time \(t\) in the \(xy\) plane is given by \((x,y) = (t - \sin t, 1 - \cos t)\). What is the distance traveled by \(P\) in the interval \(0 \leq t \leq 2\pi\)?

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